Charts and Graphs
AN INTRODUCTION TO GRAPHIC METHODS
IN THE
CONTROL AND ANALYSIS OF STATISTICS
By
KARL G. KARSTEN, B. A. (Oxon.)
CONSULTING STATISTICIAN
INTRODUCTION BY CARL SNYDER
CHIEF STATISTICIAN OF THE FEDERAL RESERVE BANK OF NEW YORK
may he laid down as almost a fundamental principle
that the statistician who is to be successful in business must
cultivate the graphic methods*^ — Leonard Ayers*
New York
PRENTICE-HALL, INC
1925
Copyrighted, 1923, by
PRENTICE-HALL, INC
All rights reserved
First printing, October, 1923.
Second printing, January, 1925.
PRINTED IN THE UNITED STATES OF AMERICA
To
E. D. K. and E. C. K.
In inadequate acknowledgment.
PREFACE
In its general structure, this book follows a philosophic,
and not an encyclopedic, arrangement. It therefore inci-
dentally supports the author’s theory of a system of natural
evolution of charts, in accordance with which all chart-forms
fall into line with simple origins and clear channels of growth.
In the light of this theory, there is no baffling heterogeneity, no
confusion of purposes or principles, in all the immense multi-
tude of existing graphic forms. On the contrary, that multi-
tude resolves itself into a consistent, organic body of simple
root-forms and logical combinations and developments. Not
only can we allocate each form to its proper place in such a
system, but we can often discover gaps in the system, and bring
to light forms which, while not yet in use, have reason to be.
As examples of such experience, the following more or less
original methods may be mentioned.
The author is indebted to a host of friends and associates,
and therefore cannot, in a wide sense, claim originality for
any of the methods described. But, in so far as no plain trail
leads from some of them to any individuals, and, in so far as
their first use is believed to be made either in this book or in
earlier work of the author, he may take some small responsi-
bility for the summary-chart, the double-probabilities paper,
the population-map, and the collapsible bead-map. The theory
of the silhouette-bar curves, and the argument for the reversal
of axes of ogives fall into the same class, as does the use of
square-root paper for economic data, and the use of the names
*‘amount-of-change” and ^^rate-of-change.^’ The same state-
ment holds true regarding the compounded-average-seasonal
method. Needless to say, these are, almost all, inevitable
results of the application of the theory that chart-forms are
naturally and logically evolved, one from another.
The greatest contribution to chart-making, from any single
source, is the Gantt Progress Chart. This chart is, unquestion-
ably, the most powerful graphic device for business and for all
vii
PREFACE
viii
executive and managerial purposes. While the description has
been rather full, as given herein, it is by no means complete;
and the Gantt charting methods, in all their co-ordinated
ramifications, constitute an independent system of accounting
and of executive control, which goes far beyond the proper
field of this book. The present volume must, therefore, be
supplemented by another, Mr. Wallace Clark’s “The Gantt
Charts,” to get the full benefits of the method. Mr. Clark’s
achievements in industrial engineering and the promotion of
managerial efficiency are ample recommendation for his book.
And the chart which has been so unqualifiedly praised and
adopted by Mr. Fred J. Miller, former President of the Ameri-
can Society of Mechanical Engineers, by Mr. Walter N.
Polakov, a leading authority on power engineering, and by
European experts, needs no endorsement from statisticians.
Inadequate mention has been made in the text, of the work
of Professor William F. Ogburn (“Social Change”) on the
geometric trend of human culture and civilization, which has
gone far to influence the presentwriter in his presentation of the
law of organic growth as the great raison-d’ etre of rate-of-change
curves. Professor Ogburn’s careful and keen pioneer work
in this field will have an increasing effect upon economic
thought for a long time in the future.
In a different field, the work of Mr. Carl Snyder should be
referred to in any discussion of chart methods as it has set a
high standard in statistical research, and has, by graphic inter-
pretation, given to abstruse economics a vital and practical
bearing upon business and commercial welfare. He has been
a leader in bringing mathematical skill, economic research, and
business problems together. The student of chart-making can
do no better than to study the methods used in the charts
appearing in the Monthly Review of the New York Federal
Reserve Bank, from which, as will be seen, we have drawn
heavily for illustrative material.
Fewer, but as excellent, are the charts appearing in the
Bulletin of the Cleveland Trust Company, prepared under
the direction of Dr. Leonard P. Ayers. These charts, and
Mr. Snyder’s, are models. The charts of the Harvard Bureau
of Economics, though of a single type, are always powerful
and well-made._ The charts in the Monthly Survey of Current
Business^ published by the Department of Commerce, are
also well-drawn. Indeed, the use of good charts is steadily
PREFACE
IX
increasing. We have seen few books so well illustrated with
excellent charts as Dr. Ayers’ “The War with Germany,” or
Mr. Joseph E. Pogue’s “Economics of Petroleum.”
Others to whom acknowledgment should be made, not alone
for contributions to this particular volume, but for important
contributions to the growth of a sound and elEcient charting-
practice, are Mr. John Wenzel and Mr. Arthur R. Burnet,
both of whom, with the author, earlier enjoyed the privilege of
working with that pioneer in the field, Mr. Willard C. Brinton.
To many other economists and former associates, among whom
may be mentioned Professor Robert E. Hale, Professor Paul
Douglas, Mr. Stuart Chase, Mr. Paul Brissenden, Dr. Fred R.
Macauly, Mr. John Scoville, and Mr. Richard Webster,
the author is indebted in innumerable ways. The courteous
permission of authors of books and articles in the same field,
to borrow illustrations from their works, is appreciated, and
the attempt has been made invariably to credit the sources of
such illustrations as they appear in the text. It is a pleasure
and a duty to recommend such important books as those of
Lipka, Peddle, Running, Haskell, and Brinton; also the
statistical treatises of Yule, Bowley, Secrist, King, and Kelley.
A word may be said as to the style of the text. It is a
quaint and curious folk-way of the academic world that a
technical account is worthy of respect directly in so far as it
can not be understood. This hoary tradition is not limited
to college walls — ^the rocky road to business, until recently,
has rested on the self-same supposititious secrecy, and the paths
of all professions lead to inner circles that guard, as best they
can, the knowledge and the standards of their work. When
such precautions make for better craftsmanship, they are most
heartily to be endorsed. But, when they merely further
selfish ends, they are a plague and pestilence, and those who
practice them, only that their own minute monopolies of craft
may be entrenched, come, sooner or later, into the class of
parasites, retarding the growth of their profession.
Having confessed so little patience with the doctrine of
the incomprehensible per se, we have naturally sought to
empty the entire bag of tricks, and to tell the whole story
of the chart in the simplest words that we command. Our
belief has been that it is a lesser sin to be too easily understood
than never understood at all. But at the same time, we
have sought to make the story full and complete. If any of
X
PREFACE
our readers find charts which do not fall into place in this
account, but would appear to have been omitted, we beg that
they will freely advise and assist us to include them. It is,
moreover, probable that, in spite of vigilant revision, many
errors have crept in; we hope that readers who detect them will
courteously co-operate by sending corrections, suggestions,
and criticisms.
Chart-making is an art which all can practice. But there
will always be a world of difference between the charts of
amateurs and those of master-statisticians. Perhaps the day
is not far off when, from the latter class, will come a group
collectively intent on keeping up the standards, not the secrecy,
of graphs and all statistics. The need for some criterion, high,
but not too high to be effective, has been already felt, and
efforts to establish safe statistical standards are on foot. It
is our understanding that we may shortly look for sets of
standard texts and examinations, from a committee of the
American Statistical Association, under the chairmanship of
Mr. Malcolm C. Rorty. Such steps will be warmly welcomed
in the profession. The task is to set good standards and to
make them public property in good plain everyday English.
And as a contribution to the protection of the calibre of busi-
ness statistics through the medium of graphic presentation,
this book is offered.
New York City, September, 1923
Karl G. Karsten
TABLE OF CONTENTS
Page
Introduction by Carl Snyder xxxvii
BOOK I. SIMPLE CHARTS
Part I. Non-Mathematical Charts
C hapter Page
1. Maps and Diagrams 1
IL Classification-Charts 13
III. Route-Charts 21
IV. Composite Charts 39
Part IL Amount-of-Change Analysis
V. Statistics 48
VI. Work-sheets S3
VII. Co-ordinates 63
VIII. Dimensions and Variables 74
IX. Hundred-Per-Cent Bars 83
X. Pie-Charts 89
XL Bar-Charts 99
XII. Composite Bar-Charts Ill
XIII. Pictorial Bar-Charts 124
XIV. Vertical Bar-Charts 134
XV. Curves 145
XVI. Fields 154
XVII. Scales 167
XVIII. Plotting-Points 190
XIX. Composite Curves 198
XX. Historical Curves 220
XXL Cycles 235
XXII. Zee-Charts '. 252
XXIIL Progress-Charts 261
XXIV. Summary-Charts 278
xii CONTENTS
XXV. Silhouette Bar-Charts 285
XXVI. Index Numbers 294
XXVIL Frequency Series 308
XXVIII. Frequency Curves 326
XXIX. Ogives ■ 341
XXX. Lorenz Curves. 356
BOOK II. ADVANCED CHARTS
Part III. Rate-of-Change Analysis
Chapter Page
XXXI. The Genealogy of Numbers 366
XXXII. The Law of Organic Growth 377
XXXIII. Rate-of-Change Analysis *. 382
XXXIV. Rate-of-Change Scales 387
XXXV. Rate-of-Change Curves ■ . 402
XXXVI. Historical Rate-of-Change Curves 416
XXXVII. Logarithmic Frequency Curves 426
XXXVIII. Logarithmic Ogives 444
Part IV. Special Analyses
XXXIX. The Normal Curve of Error 450
XL. Probability Curves 454
XLI. Shifted Zero-points 472
XLII. Curve Fitting 477
XLIII. Specially Projected Scales 482
XLIV. Formulae for Curves 490
Part V. Calculating Charts
XLV. Curves for Formulae 511
XLVI. Parallel Nomographs 533
XL VII. Zigzag and Composite Nomographs 560
XLVIII. Slide-Rules 577
XLIX. Hundred-Per-Cent Triangles 588
CONTENTS
xiii
Part VI. Two- and’ Three-Dimension Data
L. Hundred-Per-Cent Squares 598
LI. Area-Bar-Charts 613
LII. Population Maps 623
LIII. Models 630
LIV. The Third Dimension 634
LV. Frequency Surfaces 650
LVI. Relief Maps 661
Part VII. Conclusion
LVII. The Statistical Materials 671
LVIIL The Function of Charts 684
Appendices
Appendix A. Implements for Making Charts 691
Appendix B. Steps in Making Charts 696
Appendix C. Methods of Presenting Charts 702
Appendix D. Colors in Charts 709
Appendix E. Optical Illusions in Charts 711
Appendix F. The Verbal Chart 713
Bibliography
Short Bibliography 715
Indices
Index of Persons and Sources 717
Index of Illustrations by Subject Matter 719
General Index 725
LIST OF ILLUSTRATIONS
By Chaut-method
FIG.
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PAGE
Pictorial Map of the United States
flEARt-SHAFH) M.AF OF THE WORLD
Meecai<jr'.s Projection of the World
Hemisphi-rical Projection of the World
Klliftr’al Projection of the World ’
Homolooraphic Projection of the World
Data for a Floor-flan.
Samples of Cross-ruled Paper
An Unfinished Floor-plan
The Floor-plan, Finished
The Nomenciatcrk of Co-ordinates
A Simple Box-chart
Chart with Boxes of Various Shapes
Radiating ok Planetary Chart
An Example of Complicated Data
Five: Interlocking Classification Charts —
Tree-chart.
A Tabilation of Simple Route-chart Data
A Condensed Work-sheet
A Very Condensed Work-sheet
The Simplest Procedure-chart
A Simple Procedure-chart with Many Items
A Pictorial Procedure-chart
A CyRAFtuc Outline of Thought
A Popular Presentation
An Excellent Pictorial Route-chart
The Analogy of Vats, Tanks or Reservoirs
A Simplified (iilbrkth Process-chart
A Simple Form
A 'Fimh Record, But Not a Time-chart
A Cantt Chart. ... — . —
A Simple Time-chart, *
A Weekly Clock-chart
An Annual Clock-chart
Route-map.
Fin-map -
An Isometric Drawing
Routing on a Classification-chart
Classification-chart and Map
A Simple Computing Sheet
Various Geographic Groupings of the States
An Incomplete State-list Geographically Arranged
Classified Headings to the Columns
Column Symbols, Formulae, and Classified Stubs and Captions . .
Column Symbols and Computing Instructions
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XV
xvi . LIST OF ILLUSTRATIONS
FIG. PAGE
46. Steps in Layout of Co-ordinates 63
47. Same 64
48. Same 66
49. Same 67
50. Field with Equal Scales 68
51. Scales of Axes Unequal 68
52. Origin of Chart near One Edge of Field 70
53. Origin in Corner of Field 70
54. Origin Not Shown in Chart. . 71
55. Field with Co-ordinates Not Perpendicular 71
56. The Three Axes of Three-dimensional System of Perpendicular
Co-ordinates 72
57. Polar Co-ordinates 72
58. Two Lines or Bars, One Twice as Long as the Other 78
59. The Area of the Second Square is Four Times the First 78
60. The Area of the Second Square is Twice That of the First. ... 79
61. The Area of the Second Circle is Four Times the First 79
62. The Area of the Second Circle is Twice the First. 80
63. The Height of the Second Figure is Twice the First 80
64. The Heights of the Two Figures are in the Ratio of One to the
Square Root of Two 80
,65. The Effects of Comparison by Linear, Square, or Cubic Measures 81
66. A Simple 100% Bar 83
67. Many Segments, No Shading 84
68. Classification-chart and 100% Bar 85
69. Distinct Shading 86
70. Two Bars are Easily Compared 86
71. Comparison of Three Different Years 87
72. A Simple 100% Circle 90
73. Accurate Comparisons Cannot be Made 91
74. The Less Detail, the Better 92
75. Labelling is Difficult 93
76. Excellent Dollar Charts 94
77. Shading the Segments to Increase Popularity 95
78. How to Calibratb the Circle 96
79. Many Segments, No Shading 97
80. A Pie-chart IN’ Metal 97
81. A Simple Bar-chart. 99
82. Detailed Data may be Included 100
83. A Long Bar Broken to Save Space 101
84. National Distribution by States and State-groups 102
85. An Alphabetic Arrangement 103
86. Historical Data Must be in Order 104
87. Placing thh Most Important First 105
88. The Arrangement in Order of Size is Popular 106
89. Thb Gantt Idleness-chart 106
90. Classification-chart and Bar-chart 107
91. Thb Total Bar is Wider 108
92. An Office Record Form to Include Bars 108
93. Bars as Part of the Office Record 109
94. Typewritten Bars for Typed MS 110
95. The Compound Bar-chart Ill
96. The Chart Does Not Suffer from Detailed Statistics Attached 112
97. Very Small Segments May Be Shown 113
LIST OF ILLUSTRATIONS xvii
HG. page
98. The Relative (or Percentage) Bar-chart 114
99. Any Pair of 100% Bars Really Forms a Relative Bar-chart.... 114
100. The Multiple Bar-chart 115
101. A Good Comparison of Historical Data 115
102. Correlation is Indicated by Mirroring 116
103. Symmetry has Only a Popular Value 117
104. Connection Lines or Shadings to Distinguish Segments 118
105. Warping the Chart to Show the Treni> of Changes 119
106. Connection Lines. (Note Inserted Data) 120
107. A Compound Multiple (Absolute) Bar-chart 121
108. A Compound Multiple (Relative) Bar-charT 122
109. The Compound Relative is the Best of the Composite Bar-charts. . 122
110. The Simpler Forms Are More Effective 123
111. The Circles Must Have Uniform Radii.. 126
112. Segmented, Like the Compound Bar-chart 127
113. Suggesting Metal Coins 127
114. Pictorial Figures May be Substituted for Bars 128
115. The Third Dimension is Ornamental # 129
116. One of Many Devices to Stimulate Interest 130
117. Pointers, Instead of Segments, Suggest Pressure Guage Dials. . . . 131
118. Aeroplanes, Horse-races, Boatraces, and the Like, Have a Certain
Popular Value 132
119. A Vertical Bar-chart 134
120. A Series of Vertical Bars 135
121. Bar-chart with a Field 136
122. The Vertical Bar-chart with Data Reading Upward 137
123. The Data is More Easily Read 138
124. A Relative Multiple Bar-chart 139
125. Note the Key to the Shading Accompanying the Data 140
126. An Absolute Multiple Bar-chart 141
127. Wide Bars with Data Inserted 142
128. Connecting Lines are Often Useful 143
129. Connecting Lines Used in Comparison of Two Different Years. . 144
130. Here is the Data — Historical 145
131. The Ordinary Bar-chart 146
132. Vertical Bars for Popularity 146
133. A Curve Through the Bars 147
134. The Bars Disappearing; the *Tield” Appearing 147
135. The Evolution of the Curve is Complete 148
136. Here is Data not in Series 150
137. No Curve Should be Made with This 151
138. The Amputated Chart is Deceptive 154
139. The Case Against Amputation is Clear 155
140. When Zero is Arbitrary, It can be Omitted 156
141. The White Zone Warns the Reader 158
142. The Uneven Base-line Indicates That it is Not the Real Base-line 159
143. A Wavy Base-line is a Short-hand Warning 159
144. The Zero-line Should Always be Heavy 160
145. The Sound Position for Two Vertical Scales 162
146. An Interesting Comparison of Different Periods 163
147. The Vertical and Horizontal Scales are Equal 167
148. The Horizontal Scale is Twice the Length of the Vertical Scale 168
149. A Curve on a Field with Equal Scales 168
xviii LIST OF ILLUSTRATIONS
• FIG. . page
150. A Curve on a Field with the Horizontal Scale Twice the Length
OF the Vertical Scale ‘ .. 169
151. The Vertical Scale is Twice the Length of the Horizontal Scale . , 169
152. A Curve on the Field with the Long Vertical Scale 170
153. A Comparison of Curves drawn on Different Vertical Scales,. . 171
154. Examples of Convenient Horizontal Scales 173
155. Showing One Month by Days on Letter-size Paper 174
156. One Year by Months on Letter-size Paper 175
157. One Decade by Years 176
158. One Quarter-century by Years 177
159. One Year by Weeks on Double Letter-size Paper 178
160. One Year by Weeks on Letter-size Paper 179
161. Several Charts Overlaid Appear as One 180
162. The Freak Peak Need Not be Accommodated 181
163. Data to be Charted 182
164. The Highest Point in the Series Should be Plotted about Two-
thirds UP THE Chart-field 183
165. Commercial Forms Available 184
166. To Obtain a Scale Smaller Than Those Given By the Ruler 185
167. Engineers’' Triangular Rule .86
168. To Obtain a Scale Larger Than Those Given by the Ruler 186
169. Examples of Convenient Vertical Scales 187
170. Table for Vertical Scales with Engineers’ Rules on 6-, 8-, and
10-INCH Fields 188
171. A Chart-field 190
172. To Plot Anywhere Between Ordinates 192
173. To Plot Only Upon Ordinates 193
174. Data With Different Intervals 194
175. Interpolation for the Period of the War and Extrapolation for the
Years after 1919 196
176. Extrapolation 197
177. Each Curve Has Its Own Vertical Scale 198
178. Each Curve Has Its Own Horizontal Scale 199
179. The Heavy Line is Used for the More Important Curve 200
180. Zones Instead OF Curves 201
181. An Excellent Form of Zone-curve 202
182. It is Useless to Show All the Individual Curves 203
183. An Excellent Adaptation of the Zone-curve 204
184. A Gun-shot Chart 205
185. The Staircase Curve is Near to a Bar-chart 206
186. An Absolute Compound Pipe-organ Bar-chart or an Absolute
Staircased Band-chart .... . ..207
187. The Smoothed .and Staircase Curves Differ in Outline and Areas 208
188. Pseudo-staircased Curve 209
189. Several Pseudo-staircased Curves 210
190. A Pseudo-staircased Curve 211
191. An Interesting Use of Shadings in a Band-chart or Vertical Bar-
chart 211
192. The Curves are true Only for Cumulations of the Layers 212
193. A Relative (or Percentage) Band-chart 213
194. The Smoothed Relative Band-chart 214
195. The Staircased Relative Band-chart 215
196. A Smoothed Relative Band-chart 216
197. A Smoothed Relative Band-chart . . . . . 217
LIST OF ILLUSTRATIONS xix
FIG. page
198. An Excellent Band-chart (Absolute) 218
199. The Relative Chart is Supplementary 218
200. A Pictorial Curve 219
201. A Historical Series 220
202. Year by Months, Universal Ruling 221
203. The Individual Charts Combine Easily 222
204. Fanning Up and Down to Compare Seasonals 224
205. Simple Series and Annual Cumulations 225
206. Series and Cumulation Plotted With Same Scale 226
207. This Data Cannot be Cumulated 227
208. The Simple Series and Its Moving Annual Total 228
209. Three Positions for the Same Moving Total 231
210. A Detail of the Last Figure 232
211. Moving Annual Total and Average Series 233
212. The Moving Annual Average Gives the Trend 234
213. The Mechanical Cyclograph 235
214. As a Chart, This is Worthless 236
215. The Rectilinear Co-ordinates Are Much Better 237
216. A Free-hand and Imaginary Picture of the Business Cycle 238
217. Cycles of Slightly Varying Lengths 239
218. Showing the Use of Relative (Percentage) Figures and a Rounded
Curve 241
219. Daily Cycles 242
220. Comparison of Two or More Cyclic Periods on one Chart 243
221. Cycles May Change 244
222. The Moving Average Shows Trend 245
223. The Seasonal Cycle Computed from the One Cyclic Period 247
224. The Seasonal Computed from the Trend 248
225. A Remarkable Case of Changing Cycle Fluctuations 248
226. The ^‘Compounded Average” Seasonal 250
227. A Year by Weeks 253
228. Four Zee-charts Forming a Single Series 254
229. Two Charts Fanned Out For Successive Years 256
230. Two Charts Fanned Upward to Study Seasonals 257
231. Table FOR Zee-chart Scales 258
232. Mr. Burnet’s Arrangement 260
233. Detail of a Progress-chart Form — Blank 264
234. Table of Data for a Progress-chart *. 265
235. The Quota is Entered For a Year Ahead 266
236. The Chart On January 31st 267
237. The Chart on February 28th 268
238. The Chart on March 31st 269
239. A Progress-chart of Sales by Districts 270
240. A Progress-chart Used for a National Inventory 272
241. Pencils Are Used in the Work-shop 274
242. Data on a Short Fly-sheet — ^The Gantt Way 275
243. The Flow of Goods. . . 278
244. A Typical Industrial Process 279
245. The Summary-chart 281
246. A Net-worth Chart-form 282
247. A Customary and Sound Combination of Bars and Curves 284
248. A Curve is too Detailed and Large 28i
249. The Essential Data 286
250. The Same Curve Seen from Its End 288
XX
LIST OF ILLUSTRATIONS
FIG.
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PACK
A Detailed Form 289
Data in the Chart 290
Simple Silhouette Bars Presented Horizontally 291
A Silhouette Bar-chart Set Horizontally 292
A Comparison of Two Curves on the Same Chart 296
A Comparison of Several Charts 297
Obviously, Only Index Numbers are Possible 298
Various Indices of the Same Phenomenon Produced by Different
Methods of Weighting 299
300
301
302
303
Correlation Shown by Mirroring 304
A Slightly Lagged Correlation 305
306
308
309
309
The Raw Material for a Frequency Series 311
Another Crude List 312
The First Step is Arrangement By Magnitude 313
A Common Tendency to Bunch Up 314
Piling Up on the Round Numbers 315
Comparison of Fourteen Series Derived from the Same Data by the
Use of Different Group Limits and Group Sizes 316
Comparison of Curves of Three Series Derived from the Same Data 317
The Frequency Series 319
320
321
321
322
323
Period Data 32 /
Period Data 328
Period Data 328
Point-and-Period Data 329
PoiNT-AND-PERIOD DaTA 329
Comparison of a Staircased and a Smoothed Frequency Curve... 330
The Staircased Form is Appropriate 331
A Very-simplest Staircased Curve 331
The Smoothed Form is Necessary 332
It is Difficult to Compare Two Staircase Curves 333
A Cumulable Series 334
A Non-cumulable Series 335
A Rounded Curve
Computed Averages Must be Used for the Irregular Intervals . . 337
A Zoned Frequency Curve 338
A Frequency Band-chart 339
A Relative Frequency Curve ’ . . 340
The "Tess-than” Cumulation 341
The *‘More-than’' Cumulation *342
LIST OF ILLUSTRATIONS xxi
FIG. PAGE
303. An Example of a Frequency Series (So-Called) Which Cannot be
Cumulated 343
304. Two Ogives Are Always Possible 344
305. Showing the Four Possible Cumulations For Point Data 345
306. The Four Possible Cumulations For (Point-and-) Period Data 346
307. The Rounded Ogives 347
308. Evolution of the Ogive (Staircased) 348
309. Evolution of the Ogive (Smoothed) 349
310. The Simple Curve and Its Two Ogives (Staircased) 350
311. Relative Data 351
312. Comparison of Absolute Data is Sometimes Difficult 352
313. Comparison of Relative Data is Easy 353
314. Secondary Data at the Right 354
315. The First Measure — By Count of Items 356
316. The Second Measure — By Count of Units 357
317. Both Measures 357
318. Cumulating the Percentages 358
319. Data for the Lorenz Curve *. 359
320. Data to Plot the Lorenz Curve 359
321. The Lorenz Curve 360
322. What the Lorenz Curve Tells the Layman 360
323. The Familiar Example, . . 361
324. Two Curves of the Same Data By Using Both *'More-than'' and
‘Tess-than” Cumulatives 362
325. Two Curves of Different Data 363
326. The Logical Form is Triangular 364
327. Table of Logarithms, 1-5 374
328. Table of Logarithms, 5-9 375
329. The Rate-of-change Curve — First Method 388
330. The Rate-of-change Curve — Second Method 389
331. The Rate-of-change Curve — ^Third Method 391
332. Good Rate-of-change Chart-fields 392
333. A Rate-of-change Part-deck Form 394
334. Part-deck Rate-of-change Paper 395
335. A '‘Split-deck” from 30 to 300 396
336. Note the Special Scale at the Right 398
337. One Way to Find the Rate-of-change Scale 400
338. Comparison of Series Lying in Different Parts of Chart, though
Not Fluctuating Greatly 403
339. Amount-of-change Curve (Absolute) 404
340. Amount-of-change Curve (Relative Numbers) 405
341. Rate-of-change Curve (Absolute) 406
342. The Percentage-Increase-or-Decrease Recalibration 408
343. Several Curves Drawn on the Same Scale 409
344. Several Scales in a Single Split-deck 410
345. Shifting Curves to Avoid Insignificant Crossings 412
346. An Amount-of-chan9E Chart • 414
347. A Rate-of-change Chart 415
348. Long-time, Series of Economic Data 417
349. Short-time Series of Economic Data 418
350. A Careful Positioning of the Curve 420
351. Comparison of Rate-of-change and Amount-of-change Curves... . 422
352. Interpolation and Extrapolation 423
353. Careful Extrapolation 424
xxii LIST OF ILLUSTRATIONS
FIG. PAGE
354. Compound Curves 427
355. A Frequency Series Which Appears Slightly Asymmetrical 428
356. An Asymmetrical Distribution 42^?
357. Yule’s Example of a U-shaped Distribution 430
358. Six Moderately Asymmetrical Distributions 431
359. The Independent Variable is Measured from an Arbitrary Zero
Point 432
360. A Moderately Asymmetrical Distribution which the Logarithmic
Scale Has Not Made Entirely Symmetrical 431
361. An Extremely Asymmetrical Distribution M^de Symmetrical by the
Logarithmic Projection 435
362. The Double-logarithmic Projection is Best 436
363. An Extremely Asymmetrical U-shaped Distribution Brought to a
Beautiful Symmetry by the Log-scales 436
364. Same as Previous — Historical Comparison 440
365. A Table of the Convenient, Nearly-geometric Intervals by which
THE Range Between Successive Powers of Ten May be
Divided 441
366. The J-shaped Distribution 442
367. Ogives Plotted Upon Logarithmic Horizontal Scale 445
368. An Ogive Plotted Upon Both Logarithmic Scales 447
369. The Normal-curve Ordinates of the Uncumulated Series 452
370. The Normal Ogive Ordinates of the Cumulated Series 455
371. Diagram Showing the Method of Constructing a Probabilities
Projection along the Y-axis 456
372. A Less Useful Form in Which the Independent or X-scale of the
Range is Readjusted to Straighten Out the Ogive 457
373. Commercial Probability Forms. 458
374. The Logarithmic Probabilities Projection in Use 459
375. An Arithmetical Projection of the Dependent (or Frequency) Scale 460
376. A Probabilities Projection of the Previous Chart 461
377. Symmetrical so Far as Data Obtains 462
378. Symmetrical but Distinctly not Normal 463
379. The Comparison of Ogives for Different Dates 464
380. The Ogive of the Units of Measurement of the Items (Star Bril-
liancy) IN an Incomplete Series is Straighter and More
Reliable than the Ogive of the Items (Number of Stars) 466
381. Another Example of the Two Interconvertible Frequencies for the
Same Data 457
382. Alternative Data Yielding the Lorenz Curve 468
383. The Double-Probabilities Projection Straightens Out the Lorenz
Curves When of Normal Distributions 469
384. Double-probabilities Projection for Several Lorenz Curves... , 470
385. A Historical Retrospect with Reversed Log Plotting for the
Horizontal or Time Axis 474
386. Another Example of the Same 475
387. The Four Lower Curves Fail to Straighten Out on Logarithmic
Vertical Scales 484
388. The Square-root Projection of the Vertical Scale Brings Much
Greater Regularity to the Curves of the Preceding Chart 485
389. The Two Ways of Straightening Out Semi-cycles of a Sine Curve 486
390. Showing How Closely the Cycles of One Set of Periodic Economic
Data Approach a Sine Curve Wave 488
391. The Linear Equation, y ^ ax c 49I
LIST OF ILLUSTRATIONS xxiii
FIG. page
392. The Curve of y = ax^ or log y ~ log a ^ b log x 494
393. The Curve of y = « & -f c or, log (y — c) = log a-^-b log a: 496
394. The Curves of Known Powers, y = ax^ -f c and, - ax -{■ c 498
395. The Hyperbolic Curves, y ^ c andv = — r — 500
X a + cx
396. The Hyperbola w'ith Three Constants, y = — r Vd 502
397. The Parabola, y = « -f * 506
398. Exponential or Logarithmic Curves 508
399. r=2A^+3 512
400. Z == 7 ~2J 513
401. Z = 2X+r 514
402. Z=r--2X 515
403. r=Z-2X 520
404. 7 = Z ~ 2Ar 521
405. A Simple Calculating Chart 522
406. Z = A'F 523
407. Chart for Determining Scales of Curve-charts 524
408. Z = ^ 525
r - 3
409. Z = y3T
410. Log V - (.05.Y - .2) ta n 9Z 526
411. Z= y(-^ + 2)"-+ r 527
412. A More Complicated Chart 528
413. A Simple Model of Profit-and-loss Computer 529
414. A Composite Chart With Many Scales 530
415. A Simple Combination of Logarithmic and Arithmetical Scales by
TH3 Use of a Curve 531
416. A Simple Parallel Nomograph 534
417. r = i(:v + 2) 535
418. r = X + Z 536
419. 7-.YZ 537
420. Y^X + 1 539
421. r = + Z 540
422. r = Z— 2X 540
423. The Inverted X-Scale 542
424. The Use of an Outer Scale for the Unknown Variable is Not Good 543
425 544
426. Construction of the Parallel Nomograph — 1 547
427. Construction of the Parallel Nomograph — II 550
428. Construction of the Parallel Nomograph — III 552
429. Construction of the Factorial Parallel Nomograph 555
430. Chart for Determining Size of Type 556
431. Chart for Determining Scales of Curve-charts 557
432. Parallel Nomograph Not Chartable by Formula 558
433 561
434. The Y-Scale Outside the Parallel Scales 568
435. The Y-Scale Inside the Parallel Scales 569
436. Construction of Factorial Zigzag Nomograph — Unfinished 570
437. Construction of Factorial Zigzag Nomograph — Finished 572
xxiv LIST OF ILLUSTRATIONS
FIG. PAGE
438. Chart to Construct Parallel Nomographs * 573
439. Chart to Construct Zigzag Nomographs 574
440. In Quadratic and Cubic Equations the Position of the Central
Axes Becomes Variable, and a Chart-field Takes the Place of
A Single Scale 576
441. A Stationary or Fixed Rule 578
442. A Slide Rule 579
443. The Magnifiers Increase the Accuracy of Readings 579
444. Slide Rule with Three Slides 580
445. A Circular Slide Rule — Pocket Size 581
446. A Special Circular” Slide-Rule 582
447. A Circular Slide Rule with Many Variables 584
448. The Same as the Preceding, Except that All Scales are Covered
and Seen Only Through Small Open Slots or Windows. . . . 585
449. An Arrangement of Pulleys, Wheels, and Weights, by Means of
Which the Pointers Come to Rest at the Roots of the
Equation 586
450 588
451 590
452. The Hundred-per-cent Triangle for Food Values 591
453 592
454, The Factorial 100% Triangle 593
455. A Single Scale Used for Two Axes 594
456 594
457 595
458 596
459 596
460. The Original Data for a 100% Square 600
461. In This Form the Data is Not Chartable 600
462. Here Each Row Totals 100% 601
463. Here Each Column Totals 100% 602
464. The Primary Division Alone Plotted from Fig. 462 603
465. The Completed Square 604
466. Here the Primary Division is the Horizontal One, Plotted from
Fig. 463 605
467. A 100% Rectangle 606
468. A 100% Square 607
469. Another 100% Square 608
470. Same as the Last in Circular Form 609
471. Anther 100% Circle 610
472. A Third Classification has been Added Here By Diagonal Divisions
and Shadings, Showing Sex 611
473. A Simple and Excellent Area Bar-chart 615
474. Vertical Area-Bars 616
475. Compound Area-Bars 618
476. Balance or Counter-poise Chart with Two Factors 620
477. A More Pictorial Form of the Preceding Chart 621
478. Every Map is an Area Chart. On this Map the Areas Represent
Square Miles 624
479. The Usual Map of the United States 626
480. On this Map the Areas Represent Inhabitants 627
481. A Plaster-of-Paris Model 634
482. A Collapsible Model 636
483. An Axono^ietric Chart (Not Isometric) 640
LIST OF ILLUSTRATIONS
XXV
FIG.
484 .
485 .
486 .
487 .
488 .
489 .
490 .
491 .
492 .
493 .
494 .
495 .
496 .
497 .
498 .
Instructions for Axonometric Chart Scales
The Wrong Way
Somewhat Better
Smoothed Frequency Surface
Staircased Frequency Surface
A Solid Model — Rounded
An Orthographic Model
A 100% Triangle Model — Four Variables
The Normal Frequency Surface — Rounded
Cross-hatched Map on the Population Projection
City Traffic Map
Floor-plan for a Small Statistical Department .
The Payzant Lettering Pen
Optical Illusions.
PAGE
, 641
, 645
646
650
653
654
656
657
658
660
667
669
688
692
LIST OF ILLUSTRATIONS
By Subject-matter
FIG. PAGE
1. Pictorial Map : Natural Resources of the United States. JBy Mr, C. Fan de
Wall 2
2. Heart-shaped Map of the World. From Bartholomew^ s Atlas 3
3. Mercator's Projection of the World. From Rand, McNally lA Co 4
4. Hemispherical Projection of the World. From Randy McNally iA Co ,, ... 4
5. Elliptical Projection of the World S
6. Homolographic Projection of the World. Adapted from maps of the Hamr
mond Map Co 5
8, Samples of Cross-ruled Paper 7
10. Floor-plan of a Small Statistical Office 10
11. The Nomenclature of Co-ordinates 11
12. A Classification of the Non-mathematical Charts 14
13. The Structure of the League of Nations. By Mr, Sidney Gulick 16
14. The Structure of a Large Merchandising Organization 17
15. Joint Interests of the Big Five Packers. From the Federal Trade Commission 18
16. Interlocking Interests of the Packers. Data from the Federal Trade Com-
mission... 19
17. The Evolution of Animal Life. From Thompson's Outline of Science*^ ... 20
18-21. Stages in the Making of Curve-charts 22-25
22. Density of Population of the United States; rank of the States at census
years, 1790-1920 26
23. Chess Openings: The Evans Gambit and its immediate alternatives 27
24. Diagrammatic Logic of the Gantt Charts. By Mr. Walter N. Polakov . ... 28
25. Routing and Channels of Sales Efforts. By Mr. Richard Webster 29
26. Flow of Supplies in the American Expeditionary Force. By Mr. Malcolm
C. Rorty 30
27. The Round Flow of Money: income and expenditure. By Mr. Malcolm
C, Rorty 31
28. Process-chart of the Loading of VB Rifle Grenades. By F. B. and L, M,
Gilbreth 32
31. An Analysis of the Stock Inventory. From Clarli s'^ The Gantt Chart N ... 35
32. Scheduling 8-hour Turns with 24 Hours Off after Six Days. From the
Bureau of Labor Statistics v * * * *
33. The Weekly Cycle of Sales in Department Stores 37
34. The Annual Cycle of Sales in Department Stores 38
35. The Salesman’s Routing on a String-map. From Rand, McNally iA Co 40
36. The Analysis of Sales at Local Branches, by map-tacks. From Randy
McNally lA Co 41
37. Routing the Work on a Statistical Report 44
38. Flow of Goods and Money through a Large Merchandising Organisation. . 45
39. Map of Garden-planting Times in the United States 46
40. Distribution of Metal Money in the World; approximate stocks in the chief
countries, 1918 54
41. State-groupings used by the Census, the Red Cross, and the Audit Bureau
of Circulations 56
42. Pig-iron Production in the United States by States, 1920 1 57
xxvn
xxviii LIST OF ILLUSTRATIONS
FIG. page
43. Illiteracy in the United States; illiterate percentage of each class (by age,
sex, race) for each group of States, 1920 59
44. Illiteracy by Age, Sex, and Race, in the United States, 1920 60
45. Savings Bank Statistics; number of banks and depositors; total, average,
and per-capita deposits; and ratios between banks, depositors, and popu-
lation. United States, 1820-1920 61
46-56. The Evolution of Cartesian Co-ordinates 63-72
57. Polar Co-ordinates 72
58-65. The Principles of Linear Illustration .78-81
66. Foreign Trade of the United States, 1920, Divided as to Exports and
Imports 83
67. Periodicals in the United States, 1920, Divided as to Period of Issue. Data
from N, W. Ayer ^ Son 84
68. American Casualties in the World War, 1917-1918, Divided as to Cause and
Nature 85
69. -The Family Budget, Divided as to Classes of Commodities, U. S., 1913.
Data from U. S. Bureau of Labor Statistics 86
70. Foreign Trade Gateways: ports of export and import, by specified groups.
U. S., 1920 86
71. The Family Budget, Divided as to Classes of Commodities, U. S., 1913,
•• 1920, and 1921. Data from Bureau of Labor Statistics 87
72. Imports into Russia, 1921. Data from Russian Information and Reviewy
London 90
73. World Statistics: land and sea areas; continental areas; geography of the
land; population of human races; language populations; religion popula-
tions; continental populations; continental languages 91
74. Purchasing Power of the Dollar of 1913 when Used for Food at Retail, U. S. 92
75. Retail Food Establishments, New York City, 1921 93
76. Analysis of Cost of Shoes, Shirts, and Suits of Clothing, as to Raw Material,
Labor, and Overhead Costs and Profits, of Textile Mill or Tanner, Manu-
facturer and Retailer. From the Federal Reserve Bank of New York .... 94
77. Analysis of Expenses of Retail Stores; expenses classified for department,
shoe, clothing, hardware, grocery, furniture, jewelry, and drug stores.
Data from Harvard Bureau of Business Research 95
78. How to Calibrate the Circle for a Scale of Percentages 96
79. Cost of the World War to the United States, as of July, 1921. Data from
the World Almanac \ 97
80. The “Swift Dollar”; analysis of income from sales. From Swift ^ Co 97
81. Business Failures, United States, 1920; amounts divided as to nature of
business 99
82. Density of Population of the Earth, by Continents 100
83. Presidential Campaign Expenditures, U. S., 1920, Data from Senatorial
Committee 101
84. Farm Property, by States and State-groups, in the U. S., 1920, in Value. . , 102
85. Trade Union Membership of the World, by Countries, 1919. Data from the
International Labor Office 103
86. Per-capita Public Debt (less cash in treasury), U. S., 1800-1920, by census
years 104
87. The Causes of Fires, U. S., 1915-19; value destroyed by specified causes.
Data from the National Board of Fire Underwriters y N, Y, 105
88. Religious Denominations in the United States, 1919. Data from “ Year Booh
of Churches^* . 106
89. Analysis of Soft Coal Production in Indiana, Illinois, and Ohio, 1917,
From Mr, Walter N, Polakov l06
LIST OF ILLUSTRATIONS
XXIX
FIG.
PAGE
90. The Foreign-born White Population Divided as to Country of Origin, U. S.,
1920 107
91. Average Weekly Earnings in U. S., and Cost of Living. From Federal
Reserve Bank of New York 108
93. Accident, Frequency, and Severity Rates in American Industries. Data
from Bureau of Labor Statistics • • 109
94. Fatal Accident Rate per 1000 Workers in Coal Mining, Specified Countries,
1919. Data from Bureau of Labor Statistics HO
95. Business Failures, Amount of Liabilities, U. S., 1916-1920. DatafromU. S*
Census ID
96. Direct Cost of Great War, National Debts of Chief Belligerents in 1919.
Data from E. M. Friedman, ^^International Finance.” * 117
97. Coal Reserves (Unmined) of the World, Millions of Tons, 1920 Estimates. . 113
1920. Data from U. S. Census IH
100. Foreign Trade of the U. S., Classified by Nature of Articles, 1920, ....... IIS
101. Publication of New Books, in Leading Nations, 1919-20. Data from **Le
Droit Auteur” Paris • .
102. Ratio of Gold Reserves of Central Banks to Paper Currency in Circulation
Compared with Relation of Exchange Rates to Par Value (March, 1922).
From Federal Reserve Bank of New York
103. Total Number of Immigrants Arrived in U. S., 1860-1920 117
104. Countries of Last Permanent Residence of Immigrants Arrived in U. S.,
1820-1920 ** :
105. Class Alignments of Population, U. S., 1870-1910. From data derived from
Census by A^ H. Hansen 119
106. Percentage of Imports Received from, and Percentage of Exports shipped
to, Different Continents. From Federal Reserve Bank of New York 120
107. Books Published in U. S. and England, 1920, Compared as to Subject.
Data from *^The Publishers* Weekly” N. Y. 121
108. Sex of Emigrants and Immigrants, U. S., 1917-1920. Data from Report of
U. S. Commissioner General of Immigration 122
109. Urban Population of U. S., 1920 , 122
110. Combined Exports and Imports of Leading Nations of World at Par of
Exchange
111. Highest Prices of Food at Retail (index numbers, U. S,). Data from Bureau
of Labor Statistics
112. Foreign Trade of the World, by Countries 127
113. The Ideal Philanthropic Budget, U. S., 1921. Data from Paul and Dorothy
Douglas, “What Can a Man Afford?” 127
114. A Half-Century of Progress in the U. S., 1870-1920. By Mr, C, F an de
Wall 128
115. Automobile Production, U. S., 1913-1921. Data from Nadi Automobile
Chamber of Commetce ^ 129
116. Savings of the World; per capita deposits by countries 130
117. Production of Basic Commodities, U. S., 1922. Data from Federal Reserve
Bulletin IH
1 18. The High Cost of Living, U. S., June, 1920. Data from the “Monthly Labor
Review.” IH
119. Production in the United States, by States, 119 134
120. Gold Reserves of the World, 1913 to 1921. 135
121. U. S. Production of Specified Commodities Compared with that of the
World H6
XXX LIST OF ILLUSTRATIONS
FIG. PAGE
122, Fatal Industrial Accident Rates for Specified Industries, Ur S., 1913. Data
from Bureau of Labor Statistics 137
123. Death-rates in Warfare, Shown as to Cause, War, and Country. Data from
Official U.S, Bulletin 138
124. Accident Mortality, by Age and Sex, U. S., 1910-12 139
125, Male Accident Mortality Rates, Shown by Age and Nature of Accident.. . 140
126. Foreign Financing in the U. S. and the United Kingdom. From the Federal
Reserve Bank of N. Y. 14*1
127. The Blame for Industrial Waste in Specified Industries. Data from **The
Elimination of WasteF 142
128, The Nature of Industrial Accidents, New York State, 1911-13. Data from
N. Y, State Dept, of Labor 143
129, Gross Tonnage of World Seagoing Iron and Steel Ships, 1914 and 1921.
From the Federal Reserve Bank of N, Y. 144
136-7. Imports into Russia, 1921. Datafrom Russian Information andReviewf*
London 1 150*1
138-9. The Amputated Chart. From Mr, John Wenzel, 154^5
140. Curative Effect of Diphtheria Anti-toxin, Datafrom U, S, Public Health
Service 156
141. Workeips* Output and Fatigue, in Dexterous Hand-work. Datafrom U* S,
Public Health Service 158
142. Average Prices of Liberty and Corporation Bonds and British War-loans.
From the Federal Reserve Bank of New York 159
143. Adjusted Index of the Volunge of Manufacture. From the Harvard Bureau
of Economic Research 159
144. Income of Railroads. U. S., 19201. Datafrom Interstate Commerce Com-
mission 160
145. Production of Automobiles, U. S., 1913-21. Datafrom Nadi Automobile
Chamber of Commerce 162
146. Comparison of Prices of 14 Basic Commodities during Civil War and World
War. From the FederaVResewe Bank of N,Y,. 163
147—53. Effects of Changes in Curve-chart scales 167-171
154r-60. Samples of Suitable Curve-chart Plotting-paper 173-9
161. Capital of New Incorporations. U. S., 1918-20. Datafrom N,Y, Journal
of Commerce 180
162. Fire Losses, U. S., 1875-1920. Data from the N,Y, Journal of Commerce,, . 181
163-4. United Cigar Store Co. Sales. 1921. Datafrom the Survey of Current
167. Samples of Chart-paper Published. Jrom Mr, John Wenzel • 186
174. Food prices in France, Great Britain, and the United States, 1920. Data
from the Monthly Labor Review 194
175. Trade-Union Membership of the World, 1910-19. Data from the Inter*
national Labor Office 196
176. Oil Consumption, U. S., 1911-1930. From Joseph E, Pogue's ** Economics
of Petroleum," 197
177. Production of Autorrmbiles, U. S., 1913-21. Datafrom National Automobile
Chamber of Commerce 198
178. Invention and War; comparison of patents during Civil War and World
War 199
179. Employment in the U. S. and N. Y. State, 1915-21. From the Federal
Reserve Bank of N»Y, 200
180. Prices and Volume of Sales of Stocks and Bonds in N. Y. Market. From the
Annalist.,.,, 201
LIST OF ILLUSTRATIONS
XXXI
FIG, . page
181. High, Low, and Average Rates for Commercial Paper, 1831-1920. From
the Federal Reserve Bank of N. Y. 202
182. Retail Food Prices. U. S., 1919-21. Data from the Bureau of Labor
Statistics 203
183. Stock Prices and the Call Money Rate, 1914-22. From the Standard
Statistics Co 204
184. Cost per pound of electrical machinery. From Leonard A, Doggeit 205
185. Magazine Advertising, 1913-21. Data from Printer's Ink 206
186. Size of the American Expeditionary Forces and Armies in the U. S., 1917-19.
From Mr. Leona? d Ayres 207
187. Magazine Advertising, 1913-21. Data from Printer's Ink 208
188. Common Labor Wages for 10 Hours Work, U. S. Steel Corp. From Mr.
Leonard Ayres 209
189. Open Market Interest Rates and Discount Rates of the Federal Reserve
Bank of New York, 1921. From the Federal Reserve Bank of Nezv York. 210
190. Call Loan Renewal Rate and Prime 90-day Banker’s Acceptances, at New
York. From the Federal Reserve Bank of New York 211
191. Bond Sales, 1889-1922. From Mr. Leonard Ayres 211
192. The Family Budget, U. S., 1914-21. Data from the Monthly Labor Review. 212
193. French Women-workers during the War, 1914-20. Data from the Monthly
Labor Review 213
194. Qass Alignments of the Population, U. S., 1870-1910. Data from A. IL
Hansen .... . 214
195. The Nature of Export Goods, U. S,, 1910-19 ... 215
196. Imports into the U. S., by Country of Origin, 1800-1920 216
197. Exports from the U. S., by Continent of Origin, 1800-1920 ... ... 217
198-9. Consumption of Gasolene by Classes of Uses. From Joseph E. Pogue's
^Economics of Petroleum." . . . . . 218
200. Changes in the Standard of Living 219
201-12. Capital Invested in New Incorporations, U. S., 1919-21. Data from
N. Y. Journal of Commerce .... 220-34
207. Wholesale Price of Bessemer Pig-iron, 1920-21. Data from Bureau of Labor
Statistics . 227
213-15. Seasonal Fluctuation in Building Operations, 1910-20. Data from F. W.
Dodge y Co. . . 235-7
214. A Mechanical Steam-pressure Record. From Walter N. Polakov 236
216. Retail Prices of Eggs. 1913-21. Data from the Bureau of Labor Statistics. 23^
217. The Forces of the Business Cycle. From Malcolm C. Rorty. . ... 239
218. Seasonal Virulence of Scarlet Fever. Data from the U. S. Public Health
Service.. . . . 241
219. Accidents in Manufacturing, Illinois, 1910-12. Data from U. S. Bureau of
Labor Statistics . . .... . . 242
220. Cold Storage Holdings of Eggs, U. S., 1916-1921. Data from Survey of
Current Business 243
221. Strikes and Lockouts, U. S., 1916-21. Data from U. S. Bureau of Labor
Statistics 244
222. Egg Production, U. S., 1920-21. Data from Survey of Current Business .. . 245
223-8. Capital Invested in new Incorporations. Data from N. Y. Journal of
Commerce 247—54
225. Typical Seasonal Changes in Interest Rates between 1890-1908 and 1917-
21. From the Federal Reserve Bank of N.Y. 248
227-32. Examples of the Zee-Chart. From Mr. Arthur R. Burnett 253-60
233-8. Details of the Gantt Progress Chart 264-269
xxxu
LIST OF ILLUSTRJTlOyS
FIG. PAGE
239-42. Examples of the Gantt Progress Chart. From Mr. JV allace Clarkes *^The
Gantt Chart.** 270-5
243. The Flow of Goods in an Industry 278
244. The Flow of Goods and Orders in an Individual Business Concern 279
245- The Accumulated Trade Balance in the United States, 1800-1920 281
247. Prices and Volume of Sales of Stocks and the Call-loan Rate. From the
Federal Reserve Bank of N. Y. 284
248-50. Gasoline Stocks, U. S., 1920-21. Data from U. S. Bureau of Mines . . . 285-8
249. Course of Production in Specified Industries, 1919-22. From the Survey
of Current Business ... 286
251. Commodity Stocks, U. S. 1919-22. Data from the Survey of Current Business 289
252. Retail Prices of Specified Commodities, 1917-21. Data from Bureau of
Labor Statistics 290
253- Production of Basic Commodities, in March, 1922. From the Federal
Reserve Bank of N. Y. 291
254. Wholesale Prices of Specified Commodities in March, 1922. From the
Survey of Current Business 292
255. Department-store Sales and Chain-store Sales, 1919-21. From the Federal
Reserve Bank of N. Y. . . . . 296
256. Department and Apparel, Chain and Mail order Store Sales, 1919-21.
From the Federal Reserve Bank of N.Y. 297
257. Production of Manufactured Goods. Data from Mr. E. E. Day 298
258. Wholesale-price Indices of 20 Basic Commodities, and Dept, of Labor
Index. From the Federal Reserve Bank of N. Y 299
259. Prices of Oil Stock and Petroleum. Data from Mr. Joseph E. Pogue 300
260. Wages and War; comparison of wages in Civil and World Wars. Data from
Monthly Labor Review 301
261. Wholesale Commodity Prices in England and U. S., 1790-1920. From the
Federal Reserve Bank of N. Y. 302
262. Wages, Prices and Employment, U. S , 1915-21. Data from the Monthly
Labor Review 303
263. Liability of Failures in U. S., compared with Wholesale Commodity Prices,
From the Federal Reserve Bank of N. Y. 304
264. Foreign Exchange Rates and Commodity Prices in Specified Countries.
From the Federal Reserve Bank of N. Y. 305
265. Wholesale Commodity Prices in Foreign Countries, 1915-22. From the
Federal Reserve Bank of N. Y. 306
266. City Finances; Per-capita revenue and receipts by sizes of cities 308
267. Production of Red Salmon in Alaska by Size of Containers. Data from
U. S. Bureau of Fisheries 309
268. Rents in Denmark by Number of Rooms. Data from Monthly Labor
Review 309
269. Effects of Diphtheria Antitoxin. Data from U. S. Public Health Service.. . 310
270, 272, 277, 289, 296, 308, 309. Per-capita Fire Losses, 1919. Data from the
Naf l Board of Fire Underwriters 311, 313, 319, 330, 336, 348, 349
271, 300. Output of Workers. Data from P. S. Florence 312,340
273, 274, 275, 276, 294. College Professors* Salaries. Data from the U. S. Bureau
of Education . . ^ 314, 315, 316, 317, 334
278, 283. Duration of Strikes. Data from Monthly Labor Review 320, 324
279, 282, 301, 302, 304. Size of Farms, U. S., 1920 321, 323, 341, 342, 344
280, 281. Gold Production of the World, 1493-1919 321, 322
284. Membership of Strikes. Data from Monthly Labor Review 327
285. Size of Factories, U. S., 1914 328
286. Value of Manufactured Products, U. S., 1914 328, 356-364'
LIST OF ILLUSTRATIONS
287, 306, 307. Hours of Labor, U. S., 1914 329, 346, 347
288. Economical Speeds of Trucks 329
290. 305, 310. Size of Families, British Peerage. Data from s Theory of
Statistics:^ 331, 345, 350
291. Scallop-shells Distributed as to Number of Ridges. From C. R. Davenport. 331
292. Effect of Tuberculosis upon length of life. Data from L, L Dublin 332
293. Bank Salaries, N. Y. City, 1919. Data from the Federal Reserve Bulletin. . 333
295. Stature and Weight of Children. Data from the Children's Bureau. ...... 335
297. Workmen’s Compensation Payment Delays, N. Y., Pa., and Mass. Data
from Monthly Labor Review 337
298. Ages of Husbands and Wives, Great Britain, 1901. Data from Yule’s Theory
of Statistics” 338
299. Female Accident Mortality Rates, U. S., 1910-12 339
303. Expectancy of Life for Adults without Tuberculosis. Data from L. /.
Dublin 343
311. Durationof Employment, California, 1918. Data from Paul E. Brissendon. 351
312. Wages and Hours of Women-workers, Virginia, 1920. Data from Monthly
Labor Review 352
313. Wages of Office, Sales, and Shop Workers, Ohio. Data from Industrial
Commisi ion of Ohio 353
314. Wages of Female Office Workers, Ohio, 1919. Data from Industrial Com-
mission of Ohio 354
323. The Distribution of Incomes, U. S., 1918. From the National Bureau of
Economic Research 361
327“8. Tables of the Natural Logarithms 374-5
329-31. Price of Potatoes, U. S., 1913-20. Data from Bureau of Labor Statis-
tics ' • • • • 388—91
332. Samples of Rate-of-change Chart-paper. From Mr. John Wenzel 392
334. Wholesale Prices of Electrolytic Ingot Copper. From Bureau of Labor
Statistics 395
335. Wages, Prices, and Money in Circulation. Data from Monthly Labor
Review 396
336. World’s Gold Production 398
337. One Way to Find the Rate-of-change Scale 400
338. Annual Rates of Turnover of Bank Deposits. From Federal Reserve Bank
of New York ; 403
339—41. Farm and Factory Wages. Data from Um S. Department of Agriculture
and New York State Department of Labor 404-6
343. Accident Mortality Rates 409
344. Marriage and Divorce — U. S., 1887-1916 410
345. Cultural Growth in the U. S. : periodicals published, patents issued, college
students, and library volumes, 1870-1920 412
346-7. Population, U. S., 1790-1910. From living Fisher 414-415
348. The World’s Production of Gold, Iron, Coal, and Cotton, 1800-1919 417
349. Violent-death Rates from Homicides, Suicides, Lynchings, Street-accidents,
Railroads, and Automobiles, U. S., 1900-1920. Data from Tuskegee
Institute and N. Y. C. Dept, of Health 418
350. Retail Price of all Articles of Food Combined, U. S., 1913-22. From the
Monthly Labor Review 420
352. Trade-Union Membership of the World, 1910-1919, Data from Inter-
national Labor Office 423
353, Consumption of Gasoline, U. S., 1911-1930. From Pogue’s Economics of
petroleum .” . 424
XXXIV
LIST OF ILLUSTRATIONS
FIG.
PAGE
354. VitalSuperiority of the Female, England and Wales, 1851-1910. Data from
Registrar General of England and Wales 427
355. Output of Coal Miners, U. S., 1919. Data from Ethelbert Stewart 428
356. Duration of Marriages, U. S., 1887-1906 429
357. Sky-cloudiness, Breslau. Data from Yule^s Theory of Statistics** 430
358. College Salaries, U. S., 1920. Data from U. S, Bureau of Education 431
359. Rent Increases, Washington, D. C, 1920. Data from Monthly Labor
Review 432
360. Length of Words. Data from Bowley*s Elements of Statistics** 434
361. Size of Farms, U. S., 1920 435
362. Size of Strikes, U. S., 1916-21. Data from Monthly Labor Review 436
363. Female Mortality Rates, U. S., 1910 439
364. Mortality Rates, U. S., 1901-1910 440
365. American Accident Table, 1919. Data from 0. E. Outwater 441
367. Duration of Strikes, U. S., 1916-21. Data from Monthly Labor Review 445
368. Distribution of Incomes, U. S., 1919. Data from Collector of Internal
Revenue 447
373. Samples of Probability Chart-paper. From Codex Book Company 458
374. Size of Farms, U. S., 1890-1910 459
375-6. College Salaries, U. S., 1920. Data from U. S. Bureau of Education. , .460-1
377. Output of Factories, U. S., 1904-14 462
378. Wholesale Price Changes, U. S., 1891-1913. Data from Mitchell* s **Index
Numbers of Wholesale Prices** 463
379. Duration of Strikes, U. S., 1916-21, Data from Monthly Labor Review .. . . 464
380. Star-light: number of stars of specified magnitudes 466
381. Labor Turnover, California, 1918. Data from Paul F. Brissenden 467
382-3. Output of Factories, U. S., 1904-14 468-9
384. Distribution of Incomes and Taxes, U. S., 1919. Data from Collector of
Internal Revenue 470
385. Unemployment in the World, 1913-21. Data from Monthly Labor Review. 474
386. Wholesale Prices in the World, 1913-21. Data from Monthly Labor Review. 475
387-8. The World’s Commerce, 1800-1919 484-485
390. Cold-storage Holdings of Eggs, U. S., 1916-20. Data from Survey of Cur-
rent Business 488
407. Chart for Determining the Scales for Curve-charts 524
412, Chart for Solution of Quadratic and Cubic Equations. From Joseph
Lipka*s Graphical and Mechanical Computation.** 528
414. Chart Showing Loads on Important Engine-frame Members. From E. A.
Andrews 530
415, . Chart Showing Proper Current Density for Copper Transmission Lines.
From B. B. Hood 531
431. Chart Showing the Proper Size of Type 557
432. Chart Showing Effects of Off-center Holes in Phonograph Records ....... 558
438. Chart for Determining Scales of Parallel Nomographs 573
439. Chart for Determining Scales of Zig-zag Nomographs 574
440. Chart Showing Bond-Yields. From Prentice-Hall^ Inc 576
441. Chart Showing Force and Velocity of Winds 578
442-3. Slide-rule and Magnifier. From Keufel and Esser 579
444. Slide-rule for Measurements of Beltings. From Carl G. Barth 580
445. A Circular Slide-rule, From Keufel and Esser 581
447-8. Slide-rule showing Costs of Book-printing S84-S
446. Slide-rule for Power-plant Calculations. From Walter N. Polakov. 582
452. Chart showing Fat, Protein, and Carbohydrates in Food. From Malcolm
C. Rorty 591
LIST OF ILLUSTRATIONS
XXXV
453. Settlements of Strikes, U, S., 1916-21. Data from Monthly Labor Review.. 592
460-66, 472. Occupations of the Gainfully Employed, U. S., 1920. Data from
U. S. Bureau of Labor Statistics 600-05, 611
467. Wages in Manufacturing Industries, Ohio, 1919. Data from Industrial
Commission of Ohio 606
468. Occupations of the Population, U. S., 1918 607
469-70. World’s Coal Supply (Unmined) 608-9
471. Jewish Population of the World, 1920 610
473. Wholesale Sales, 1922. From Federal Reserve Bank of New York 615
474. Average Incomes of Tax-payers. Data from Collector of Internal Revenue. . 616
475. Earnings of Corporations. Data from Collector of Internal Revenue 618
476-7. Charts Showing Equations of the Quantity Theory of Money. From
Irving Fisher 620-1
478. Map Showing Value of Farm-land, U. S 624
480. Map Showing Population of States . . . . 627
481. Model Showing Gas-mixtures for Gas-engines. From John B. Peddlers
^^Construction of Graphical Charts^^ . 634
482. Collapsible Model. From John B. Peddlers Construction of Graphical
Chart/^ 636
483. An Axonometric Model-Chart. From John B. Peddle' s Construction of
Graphical Charts" 640
484. Tables of Scales for Axonometric Charts. From John B. Peddle* s **Con*
struction of Graphical Charts" 641
486. Map Showing Distribution of Cattle, U. S 646
487,89,91. Wet and Dry Months of the Year 650, 654, 657
488. Stature of Fathers and Sons. From G. U. Yule's Theory of Statistics" . . . 653
490. Model Showing Cost of Electric Lamps. From R. E. Scott 656
492. Model Showing Efficiency of Copper-alloys. From John B. Peddle* s
** Construction of Graphical Charts'* 658
493. The Normal Frequency Surface. FromG. U. Yule's "Theory of Statistics** 660
494. Map Showing School Truancy, U. S., 1920 . . 667
495. Map Showing Density of Traffic in Chicago. From Haskell's "How to
Make and Use Graphic Charts" 669
496. Floor-plan of a Small Statistical Department 688
497. The Lettering Pen. From Keufel and Esser 692
498. Optical Illusions. From the Grolier Society^ 711
INTRODUCTION
In and since the War the use and development of charts
has been almost phenomenal — so large, indeed, that at least
one able economist who is interested in such things thinks
that we as a country have gone chart-mad. But this develop-
ment has not been confined to this country, and it has a very
solid basis in practical utility. There is little question that
the chart represents a genuine saving in time and in mental
effort.
In this it does not differ from the ordinary map. Suppose
the mariner, the shipping clerk, or the school boy had to locate
a given point on the earth with a statement, let us say, that it
was two thousand miles southwest from London, twelve hun-
dred miles south of New York, eight hundred miles north of
Rio de Janeiro, and so on. All of this information might be
useful and even, for certain purposes, necessary. It is, so to
speak, the statistical data of the question. But it yields no
picture. A map or a globe gives us this mental picture almost
in a flash. And that is precisely the use and service of a chart.
Let us take an example:
Within the last few months from this writing, the news-
papers have been filled from day to day with reports of this or
that industry making a “new high record.” The figures give
the idea of a prodigious boom, and, as we have so sadly learned
to know, practically every boom is followed by a crash. So
the wise man will shake his head at these “new high records,”
and sagely observe that “it cannot possibly last.”
Well, in most industries with which we are acquainted,
such new high records are the normal and usual thing, and the
absence of them the abnormal. In other words, practically
every industry, just like the population of the country, has a
fairly steady rate of growth, and so, with sharp interruptions
that come at more or less irregular intervals, it is the normal
and characteristic thing that they should make these new
high records. Naturally, such high records should at least
not be regarded in the light of sensational news.
xxxvli
INTRODUCTION
xxxviii
Let us take our old friend pig iron as an instance. We have
monthly records of pig iron production running back for forty
years. In twenty-four of those forty years some month of
those years has made a ‘^new high record’^ in pig iron produc-
tion, that is, in 60 per cent of the cases.
Furthermore, these new peaks of production tend to run
in sequences of four, five, and six years. So if we see an esti-
mate that pig iron production for this year, let us say, will
^^break all records, ’’ we know that this is a rather foolish way
of putting it, that it is simply the fairly normal thing and what
we might reasonably expect in the absence of any powerfully
disturbing causes like a world war or a profound depression in
trade.
Now all this information you may laboriously dig out of
the actual figures if you like, but you can get it all in a quarter
or maybe a tenth of the time if it is spread out in chart form.
Like the point on the map, all these relations there stand out
vividly and almost instantly.
But it is not alone the economy of time and effort that is
involved. The great thing, often, is that the chart will flash
the thing not merely to the eye but to the mind; I mean that
the picture gives you the idea of making the computation,
and even that there is such a thing as a normal rate of growth,
as in pig iron production. Lacking the picture, we might have
little to prompt us to make the investigation or suggest even
a hypothesis.
I know there are those to whom this easy method of mental
traveling is not attractive, and even, perchance, a little irri-
tating. Nothing else could explain, for example, why it is
that our mathematicians should often go through long and
laborious calculations in an endeavor to find out whether any
close correlations exist between two sets of data, or whether
a periodogram is going to fit a given set of figures sufficiently
to make it the basis for a forecast, when there is a far quicker
route. While recognizing to the full extent the value which
these methods may have in competent hands, it is still literally
true that thousands upon thousands of calculations of every
kind and description have been made as to these degrees of
correlation and all their like, involving hundreds and even
thousands of hours of needless and useless work, when a near
approximation in ninety per cent of the cases could generally
have been obtained with a log chart in much less than an hour.
INTRODUCTION
XXXIX
The typical mathematical bent of mind seems to luxuriate in
difficulties, long calculations, and complicated formulae.
The simple, swift, and direct seems to be foreign to its nature.
In our work at the Bank, we have had much reason to
study attentively these normal rates of growth. It is quite
astonishing to find how characteristic they are of the different
industries, and different lines of trade, and even such things
as the growth of bank deposits, money in circulation, and
numerous other fluctuations of the modern economic world.
They are so characteristic, in fact, that very often a log chart,
with the figure for the average rate of growth in, let us say,
the last twenty years, will suffice to identify the subject of
the picture without further label.
But this idea of the persistence of growth, as a kind of a
characteristic inertia in the different industries and trades, is
certainly foreign to our present ideas about business or the
thought of many economists. There are as yet few of our
business men or industrialists, for example, who are now
willing to believe that one can make a fairly good guess as
to, say, the average production of pig iron, or the average
railway traffic, or the average postal receipts for the years of
1930 - 33 . It is almost certain that few industries or few enter-
prises are now planned with any long look into the future.
There are very notable exceptions, like the American
Telephone and Telegraph Company and others that might
be mentioned, where the work of development is planned out
for years ahead. For most men, even in our large industrial
enterprises, these are pretty much matters of rule of thumb or
of year-to-year pressure. If it were not so, we should scarcely
have such violent ups and downs of production and trade, the
booms and slumps that bring such demoralization to industry
and to profits, and so much needless suffering among the wage-
earning population.
Some day we shall find a way around such stupidity, and
it is my own belief that the most accessible avenue is through
the grouping of the available data into interesting and well
conceived charts. They are the most reliable and most
stimulating instruments of education that we pos.scss.
So I think it has been a worthy service th at Mr. Karsten
has performed in writing such an encyclopr uic and exhaustive
work upon the subject. The time is right ior it, and it should
be highly useful. I do not mean to suggest by this that the
xl
INTRODUCTION
mere making of charts is the whole story, any more than the
possession of a fine hammer and a chisel makes a good carpen-
ter. But it is certain that, without good tools, the best of
artisans is badly handicapped, and I believe this is equally
true of the business man and the director of large enterprises.
He cannot but be going somewhat blindly if he does not have
at his right hand, maps and charts of his whole work, extending
years into the future, so that he may plan and anticipate in a
truly prescient way.
The rest of the story is that such scientific recording and
projecting into the future makes of business and industrial
enterprise a kind of romance in reality. Even the most
interesting of occupations gets to be a kind of humdrum
routine, if we have no long look ahead. Nothing stimulates
the imagination more than a well constructed excursion into
the future. And in business enterprises this is almost im-
possible without the intelligent use of charts.
But there is more. So prodigious have our industrial
activities as a nation become, so varied and so diversified,
that it is given to few men, even the ablest, nowadays, to
maintain any accurate and adequate idea of current business
trends and developments, and carry on their own work at the
same time. So I believe that soon our successful captain of
industry, like the captain on the great ocean liner, will have
always at his elbow a trained navigator or business pilot, who
will supply him with the material wherewith to study his
course and make his plans, and who will tell him at any given
moment just where he is at! And such a navigator will find
his most useful tool to be a first-hand working knowledge of
the different forms of charts which this book describes.
New York, 1923.
Carl Snyder.
BOOK 1. SIMPLE CHARTS
PART I. NON-MATH£MATICAL CHARTS
Chapter I
MAPS AND DIAGRAMS
It is probable that the original diagrammatician lived
many centuries ago, and it is not impossible that he was a
cartographer. A search for him would lead us back to the
days of “Captain Kidd” legend, when, judging by some
records, the word “chart” invariably connoted a faded sketch
of a lone island, dead trees, and buried treasure. It would
lead us back to that intrepid explorer, Marco Polo, whose
revisions of geography upset his contemporaries; back to the
Arabs, whose excellent charts of the skies played so large a
part in their nocturnal travels over the desert; and back to
the Phoenicians, who doubtless kept strange maps to guide
them about the Mediterranean shores and perhaps to warn
them of dangerously shrewd villages where the bargaining
was not profitable. We could not stop at the Egyptians,
four thousand years ago, whose floor-plans of the pyramids
have recently yielded up to us their secrets, nor at the Chinese
whose six-thousand-year-old maps of the heavens have con-
firmed modern astronomical calculations of star movements.
We should be carried back to prehistoric man, at least sixty
thousand years ago, some of whose drawings have been iden-
tified as diagrams of familiar constellations. In short, the
antiquity of maps is well established.
Not only are maps the oldest form of charting, but to this
day they are the most widely understood. And the subject-
matter they portray is of the greatest variety. Few, even of
those who use maps regularly, have any idea of this diversity.
Of the United States alone, there are on the market special
maps showing the natural resources, the density of the popu-
lation, the location and amount of the various crops, the chief
centers of the various industries, the lines of communication
and transportation. Some maps show political divisions,
others the physical contours, others the mineral subsoil, and
CHARTS AND GRAPHS
Fig. 1. Pictorial Map of the United States.
MAPS AND DIAGRAMS
3
still others the atmospheric conditions. Some show railroad
distances between cities, others show automobile distances.
It would be difficult to find any important phase of American
life for which somewhere a map is not being published and
marketed.
The student of maps will note that wherever large sections
of the earth are shown, the map seems to suffer a distortion of
outline, so that two maps of adjacent territories will not fit
closely together and form a single large map. He will recog-
nize that this is due to the fact that the earth is a sphere,
while the map is printed upon a flat surface. We are indebted
to one Christopher Columbus, who proved that the earth is
round, for the necessity of this distortion. The result is that
only upon globes can outlines be truly represented. All flat
maps being more compressed, as it were, in their centers, and
expanded at their edges, the outlines are consequently warped.
So, too, it follows that maps of large areas, such as the United
States, differ considerably in shape, according as the map is
an imaginary picture of the country from a position above its
southern, northern, or other parts.
In maps of the world, this distortion problem has become
From Fariholomew' s Atlas.
Fig. 2. Heart-shaped Map of the World.
, p ^ ^ j J J J — I — J — J —
Permission of Rand, McNally Co.
Fig. 3. Mercator^s Projection of the World.
by magnifying the polar regions and spreading before us the
sides of an imaginary cylinder. As the earth is not a cylinder
and its poles are not as long as its equator, but are merely
Permission of Rand, McNally Co.
Fig. 4. Hemispherical Projection of the World.
points on its surface, the amount of distortion can be seen to
increase gradually from the equator and to become infinitely
MAPS AND DIAGRAMS
S
great at the two poles. But by increasing the longitudinal or
north-and-south dimensions equally with the intersecting lat-
itudes, local outlines over small sections of the map are reason-
Fig. 5. Elliptical Projection of the World.
ably preserved. This device is called ‘^Mercator’s Projection.”^
Only when areas at unequal distances from the equator are
compared, does this map become grossly deceptive. Who can
forget his earliest impressions of Greenland being larger than
Australia, or his amazement at the size of Canada and his
wonder at the eno’rmous reaches of Alaska, as gained from his
world-map at the beginning of his school atlas ?
‘ Invented by Gerardus Mercator, a Flemish mathematician and geographer
■(1S12-1S94), in 1550.
6
CHARTS AND GRAPHS
Such projections, of course, have no uniform scale of miles,
for the inch that represents a thousand miles at the Desert of
Sahara will represent but a few miles near the North Pole.
A different form of distortion, a combined skewing and warp-
ing, takes place in the less common maps shaped either in two
circles or in one flattened circle or ellipse. The best preserv-
ation of true outlines and areas, that is, a more uniform map-
scale, is secured in a recent form of world-map called the
'‘orangepeel projection,’' while for less broken outlines the
“butterfly-map” and the homolographic projection^ may be
found useful. These are ingenious devices to keep recogniz-
able shapes, each gaining its advantages' only at the cost of
simplicity and continuity.
The problem of distortion due to the earth’s curvature
tends of course to disappear as the areas chosen for presenta-
tion on the map become smaller and smaller. In large state
maps it is still present, but the problem in county maps is
rarely seen, so that several county maps can be fitted exactly
together. Likewise a “scale of miles” holds true throughout
the map when the area is small. Township and city plans are
maps of still smaller surfaces and, indeed, the category of car*
tography^ is not complete until we include floor-plans and
diagrams of buildings and rooms, and the like. These are
familiar in the form of architects’ blue-prints and differ from
maps proper only in that they can be quickly prepared by
anyone, their subject-matter being of such limited space as to
require no professional engineering surveys. However, they
are in principle the same as maps, in that they are likewise
representations of space in the plane of the earth’s surface.
These plots, plans, and diagrams of small areas are so often
of great value that we shall here explain in detail how they
may be madco The first step, of course, is to secure the infor-
mation to be charted. Assuming that you wish to draw a
floor-plan, select some convenient point of reference, such as,
perhaps, a certain corner of the elevator-shaft, and from this
point of reference measure the distances to the various objects
you wish to show on the plan. Measure these distances not
2 The Encyclopedia Britannica, for example, lists some twenty-five different pro-
jections for maps of the world, of which the most distinctive have been here described,
® Cartography, according to the Century Dictionary, is the art or practice of
drawing maps or charts (that is, marine maps).
MAPS AND DIAGRAMS
7
directly to the objects, but along lines parallel to the sides of
the room. Thus a certain motor stands, say, fifty feet east
and ten feet north, of the corner of the elevator shaft. Take
a piece of paper on which the objects to be shown have been
listed in a column, and enter these figures beside each item, in
riati Of Small Statistical Office
East
from
outer
door
North
from
outer
door
Chair
Of
Statistician
8
6
N
n
First clerk
-2
5^
n
2nd clerk
si
n
Draftsman
-13
n
Typist
-12
2
etc.
Fig. 7. Data for a Floor-plan.
two columns. In the first column, headed East and West
enter “plus 50” beside the motor (“plus” meaning “east” and
Fiif* 8. Samples of Cross-ruled Paper.
The small numerals indicate the number of spaces per inch. Many other rulings
are published.
“minus” meaning “west” of your point of reference, the
elevator shaft). In the second column, headed North and
8
ClLJRrS AND GRAPHS
South enter “plus 10” (“plus” in this column meaning “north”
and “minus” meaning “south” of your point of reference).
To draw a floor-plan or diagram, since no distortion prob-
lems^ arise in such small areas, ordinary cross-ruled or “quad-
rille” paper may be used. Having prepared your data,® you
will next decide upon a “scale” or ratio of reduction to use in
the drawing, that is, what value or distance on the actual floor
shall be represented by each space or distance between lines on
the paper. It is important to pick a scale which is neither too
large nor too small, so that the drawing will be the right size
on the sheet. Suppose your paper is ruled in tenths of an inch
Fig. 9. An Unfinished Floor-plan.
with heavy rulings every inch, and you decide to let each
small space represent one foot on the floor, and each inch ten
feet. At some central spot on your paper where two heavy
lines cross, mark the letter ‘‘O’’ to represent your point of
origin. This point of origin on your paper corresponds to the
point of reference on your floor. Along the heavy line through
this “0” or zei'o-point, to the right mark the successive heavy
cross-lines ‘TO,” “20,” “30,” and so on to represent distances
east of the elevator, and to the left mark them successively
“ -10”, “ -20,” “ -30,” and so on to represent westward dir-
ection. Along the vertical heavy line through the zero-point
^ The distortion in very large buildings may amount to several inches difference
between horizontal distances of top and bottom floors, but does not appreciably
affect the rectilinear outlines of floors.
^The word ^*data^' is used throughout this book as a singular noun, unless it
refers distinctly to more than one body of statistics. Such a usage is not sanctioned
by the dictionaries, but is believed to be more in accordance with modern practice in
the statistical work-rooms.
MAPS AND DIAGRAMS
9
at right angles to the last, mark ofF the inches upward succes-
sively ‘^10, ^‘20” and so on to represent northward measure-
ments on the floor, and -10,’’ -20,” and so on downward
for southward measurements on the floor. After this, it is a
simple matter to locate and draw on the paper each item in
the spot corresponding to its true position on the floor. The
scale-numbers ‘^0,” ^TO,” etc., can be erased and the words
‘^Ten feet to the inch” or a short calibrated line, substituted.
Here you have all the elements of chart-making. It only
remains to observe the nomenclature. Our first step having
been to secure two sets of measurements for each object or
item, one for east-and-west distances and the other for north-
and-south distances, we may call these measurements indi-
vidually ^Values” and collectively "‘series.” Where, as in
this case, there are two values for each item, let us call one of
them the value and the other the “y” value in order to
distinguish them easily. If a third measurement or series of
values were present, it might of course be called the "V’ value.
In the present instance we have two, and, as will be seen
below, the east-and-west series has been taken as the
series and the north-and-south series the “y” series. This is
only a happen-so; we might equally well have reversed them,
but as it is, we can now write for convenience the letter
over our first column of figures and the letter “y” over the
second. So much for our data. It consists of two series of
values.
Now on the chart, the two lines crossing at the zero-point
or point of origin are called the “axes.” The horizontal one
is the “;i:-axis” and all values of the series are measured
along it, the positive ones to the right and the negative ones
to the left of the origin. Parallel lines above and below this
axis are called “abscissas” or “abscissae” and the axis itself
is therefore sometimes called the “axis of abscissas.” The
vertical axis is called the “y-axis” or “axis of ordinates.”
Along it the values of the “y” series are measured, positively
upward and negatively downward from the origin. The ver-
tical lines parallel to it are called “ordinates.” It will be
noticed that all points on an abscissa have the same value of
“y” and all points up or down an ordinate have of course the
same values of “a:”. Taken together as a criss-cross pattern
of lines, the abscissas (or horizontals) and the ordinates (or
verticals) are called the “co-ordinates” of the chart.
FLOOR.PLAN FOR A SMALL STATISTICAL DEPARTMENT'
MAPS AND DIAGRAMS
II
Abfictaaa
Abscissa
m
®
«5
C
®
ttf
c
•H
o
Ordinate
©
<s
c
'V
u
o
Abscissa
o
m
’T*
x-Axls or
Axis of A
o
0
3
1
ssclssae ^
Abscissa
Abscissa
Ordinate
Ordinate
Abscissa
COORDINATE RULING
Fig. 11. The Nomenclature of Co-ordinates.
The student will observe that every point on this paper
has two values, one along each axis, and that to identify or
locate a point both its values must be given. He will observe
that the axes cut the paper into four quarters (or quadrants),
in the upper right-hand one of which (in our diagram the
north-east quarter) both values of every point are positive,
while in the lower left-hand quarter (south-west) both values
are negative, and in the two other quarters one value is posi-
tive and the other negative. He will observe that, disregarding
plus and minus signs, at each side of either axis, the values
along the other axis always mirror themselves.
Many American cities are laid out in this checker-board
style. In New York the north-and-south roads are called
avenues and the east-and-west roads streets. In Washington
the former are designated by numbers and the latter by letters.
In both cases the house-numbers began at certain axial roads
and read away in both directions. The conception is the
same as that of the system of co-ordinates in the chart. Nor
is it changed when we place the point of origin at a comer
instead of in the center, that is, restrict the chart to one
12
CHARTS AND GRAPHS
quadrant, and thereby eliminate mirrored duplication of values
and the need of plus and minus signs. This is commonly done
in commercial maps, each map having a series of letters and
numbers about its edges, the letters on two opposite sides and
the numbers on the other two, each locating positions on one
of the two axes. In the index or list of cities on the map,
corresponding to our data, the proper combination of letters
and numbers for the map is given to enable us easily to find
any particular place. In short, the thoughtful reader will see
that the fundamentals of charting are already familiar ideas,
and will not allow a less familiar terminology of axeS; abscissae
and ordinates to confuse him.
Chatter II
CLASSIFICATION CHARTS
From the portrayal of space-relation between objects, we
turn naturally to the portrayal of idea-relations, and to the
relations of abstract ideas having no space-existence. Instead
of location on an actual surface, we wish to show position in a
more or less ideal scheme. We now deal, not with a geo-
graphical, but with a logical analysis. It is not possible to
illustrate all the uses of charts in diagrammatic logic, but the
classification chart is sufficiently suggestive.
In all chart-making, the material to be shown must be
accurately compiled before it can be charted. For an under-
standing of the classification chart, we must delve somewhat
into the mysteries of the various methods of classification and
indexing. The art of classifying calls into play the power of
visualizing a “whole” together with all its “parts.” Even in
the most exact science, it is not always easy to break up a
whole into a complete set of the distinct, mutually exclusive
parts which together exactly compose it.i A child can tell us
that the United States is a single nation (whole) composed of
forty-eight States and a District (parts), but almost everyone
will find difficulty in deciding the number of territories, pos-
sessions, and spheres of influence which also compose it.
A second problem arises when each of the parts is in turn
considered as a whole and its own parts analyzed.* Thus the
State of New York is composed of 62 counties, that of Mary-
land consists of 23 counties and one city, and the counties are
^The division of a whole into many parts is sometimes called polychotomy;
dichotomy and trichotomy are cases of division into two and three parts.
2 The decimal classification is a case of repeated subdivision in which decimal
figures (with or without the decimal point) are used as symbols or keys to the parts.
The Dewey decimal system for book libraries is a familiar example of this method and
the expansion thereof by the Brussels Institute Internationale de Bibliographic,
founded by Senator Henri La Fontaine, is the greatest achievement in classification
the world has ever known.
*3
CHARTS AND GRAPHS
variously divided into townships, boroughs, Incorporated
places, and so on. Even a child knows that a dollar is theo-
retically divided into ten dimes, each dime into ten cents, and
each cent into ten mills. But no two botanists agree in the
classification of flowers, for example, into families (wholes),
genera (parts), and species (sub-parts), not to mention the
elaborate hierarchies of orders, classes and divisions, and the
multitude of sub-species, sub-sub-species and hybrids.
The classification chart clearly presents, however, just so
much of this marshalling and regimentation of ideas and objects
as its author has clearly in mind. It is a method of presenting
his scheme of things instantly and interestingly. Let us
assume that he has settled his classification, has reduced it to
writing, and tabulated it with indented margins or some other
device to make it clear, and let us proceed to the technique of
its charting.
The simplest form of chart showing a whole and its parts
and sub-parts is the box-chart. It is composed of squares,
rectangles, circles, or other “boxes” arranged in serried ranks
down its page. Across the top, a single very large box carries
the name of the total group (whole). In a row beneath it and
Fig. 12. A Simple Box-Chart.
CLASSIFICATION CHARTS
^5
tied to it by connecting lines are several smaller boxes, each
bearing the name of one of the primary subdivisions (part or
sub-total).^ Beneath these again is a row of still smaller boxes,
each similarly connected to one of the boxes in the row above
and labelled with the name of one of the secondary subdivisions.
The process may be continued indefinitely downward, to sub-
divisions of lower and lower rank.
Sometimes there are so many subdivisions that they cannot
all be shown side by side. In this case there are three courses
open to us. The method most frequently employed happens
to be the least desirable. It consists in dropping some of the
minor boxes down to lower levels and connecting them vertic-
ally with the boxes above them. The method is unsatisfactory
because it complicates the reading of the chart, changing the
significance of a lower positioning on the page. When this
method must be employed, it is well to distinguish the different
ranks by various shapes, sizes or colors of boxes.
A second method is preferable. It consists in using very
deep and narrow boxes for the minor subdivisions which must
be crowded together. The labels will, of course, have to be
written downward in these boxes; they can, however, be hung
diagonally so as to make the reading easier. A third method
is an outgrowth of the second. In it the entire chart is thrown
over upon its side. The main or total box now appears at
the left of the page instead of at the top: and the process of
subdividing is carried out to the right, each rank in a different
column.4 This method is limited to cases of few ranks. Both
the second and third methods are sound in principle, the sig-
nificance of relative positioning being adhered to throughout.
The square or rectangular type of box is the best, being the
easiest to draw and the clearest to read. Often it is perfectly
feasible to omit the boxes entirely, taking care to keep the
printing in box-formation. Where two or more distinct classes
of objects are thrown together in a single chart, such as persons
and departments, it is a happy thought to give one shape of
box, such as a circle for persons, to one type of object, and a
totally different shape of box, such as a square for departments
2 The word “sub-total'* is here used* in its strict sense as an inferior or subordinate
total, a part of the grand total which can itself be viewed as a whole and split into
parts. It is not used in the customary accounting sense of a cumulative.
^ If a mathematical chart showing the value of each of the final subdivisions is
desired, the bar-charts described in a subsequent chapter may be used in conjunction
with a classification-chart in this form.
i6
CHARTS AND GRAPHS
Tig:. 13* Chart with Boxes of Various Shapes.
to the other type. Such differentiations should have a definite
purpose, however, and must not be introduced merely to em-
bellish the chart, as they then invariably complicate its reading.
Connecting lines may be either curved, straight, or rectil-
inear. The last are usually by far the best, especially where
the boxes are rectangular. Straight lines, running directly
between the boxes, give a radiating effect and are sometimes
good when the boxes are circular.* Curved lines generally fall
into the class of pointless and undesirable embellishments, but
are occasionally useful in complicated charts to connect boxes
across other connecting lines. Ordinarily, when the lines
cross, small semi-circles at the intersection on one line suffice.
CLASSIFICATION CHARTS
17
In drawing the boxes and connections, full continuous lines
will naturally be used, but it sometimes happens that certain
parts of the chart are only remotely related to the main body
of the chart, or perhaps belong to a different period of time.
In this class, fall contemplated future additions to the existing
scheme. Here the use of broken or dotted or even wavy lines
is of value, not only for connection-lines, but also for box out-
lines. Another means of differentiation, discussed later, is the
use of color or shading. It is somewhat more diverting to the
Showing the Structure of a large Merchandizing Organization.
CHARTS AND GRAPHS
CHART
JOINT INTERESTS OE
THE BIG FIVE PACKERS
Sisedon ijjvneifjli.p (“Kci-pf -nlhf ti’t ol Binks jni) fijilfOa*
■f'x/-'-" X-.*
\s N ‘ A''. /•
\'. ‘''V
A; //'A'./
P5S<::
■.(TC;-— I .(/
•my
I SI F
X ' t”'"’' ii*"' >/ i
•)I VcTi 'ntcrett of Jt*(I ptrtenl
•II Mold liitfff .1 olZl 3 H ftt eeot
*^i3 V« cppBiilc pjijf
* S OK''C(ltylntt«nj|tonilP«oiJ.cl4Co
inwhiih Atmo.r^Wi'son'SuijfCi^ir^
Reprinted from ‘'Summary Report of the Federal Trade Commission on the nieai-packine. indtisirv,
Jtdy S, 19 is:'
Fig, 15. An Example of Complicated Data.
reader, a dubious advantage, however, and it cannot well be
reproduced.
The variations of the box-chart which are occasionally seen
CLASSIFICATION CHARTS
19
JOINT INTERESTS
OF THE
“BIG FIVE” PACKERS
{Source: ^‘Summary Report of the Federal
Trade Commission on the meat-packing
industry July 5, 1918.*')
KEY TO THE CHART
Bank
Canning Company
Land Development Company
Packing Equipment Company
Cattle loan company
Miscellaneous
Rendering company
Cotton oil company
Publishing house
Railroad
Slaughtering company
Terminal railroads and facilities at
stockyards ^
Public service companies
Coldstorage and warehousing com-
panies
Stockyards companies
Fig. 16. Five Interlocking Classification Charts.
are usually inspired solely by a desire for an artistic appearance
and raise in turn some doubts about the material they present,
that is, doubts as to its accuracy or the spirit in which the chart
was compiled. The “tree-chart” is a sample of this, in which
the trunk of the tree represents the total group, the branches
the primary subdivisions, and the leaves, twigs, or fruit the
20
Cl U RTS JND GRAPHS
minor subdivisions. Another is the “planetary chart,” in
which a central sun is labelled the whole group, its planets the
primary subdivisions, and their satellites, either encircling
them or outside of them, the minor subdivisions. Such varia-
tions are justified only when something in the nature of the
object shown suggests a particular fitness in the figure. The
variation always makes for a certain amount of difficulty in
From Thompson’s **Outhne tT leme,” published by G. P. Ptibtmr's Sims.
Fig 17. Tree Chart.
Showinc; the Kvoliirion of Animal Lift*.
reading the chart, it takes a great deal more time to prepare,
and, if well done, is likely to draw more attention to its own
arrangement than to the subject-matter it is intended to
convey. ^
*The student will detect in classification-charts certain elements in common with
the diagrams described in the previous chapter. Axes are no less present because
they are not drawn and calibrated; for in one direction the positioning signifies lower
subdivision, while in the cross-wise direction it signifies equal and independent im-
portance. No scale is used because the variables are not numerical measures, but
only idealogical relations. While the analogy* is not important, it is interesting to
keep in mind.
Chapter III
ROUTE-CHARTS
To the executive type of mind few charts make such instant
appeal as those describing movement — that flow of goods
through a sequence of operations which is the keystone of
industiy. Economics itself is but the study of the successive
forms of “wealth’" through the processes of production and
distribution. Static relations, either physical, as shown in
maps, or logical, as in classification charts, may engross the
academic interest; indeed a correct conception of them is essen-
tial. But when through them is woven the added element of
time and motion, the result is lifted out of the field of cut-and-
dried research and given the values of life itself. And he who
weaves such a pattern performs, no matter in how small a
way, a creative engineering function. The picture of such a
process we call the route-chart. This chart throws powerful
light on the weaknesses and advantages of a process, either
existing or contemplated, and gives to then-eader, perhaps even
to the author, a grasp of the subject which no amount of text
can equal. It is a photograph capturing that highest of human
achievements, the mental visualization of action.
As is often the case in chart-making, preparing the data
for this chart is no small part of the work. The data consist
of the accurate record of the steps, changes or events which
take place. This record may be compiled in the form of notes
or text. In simple case's the successive steps may be listed or
tabulated, using indented margins where the process branches
or splits into different channels. Such data are very, similar
in form to the data for classification charts, already described.
But where the process is complicated with detours, by-products,
cross-connections and detailed assemblings, no list or tabula-
tion will remain clear and the data must take the form of a
careful statement in notes, possibly in conjunction with card-
indices, a cross-reference system and rough working sketches.
CHARTS AND GRAPHS
It is in fact not a bad practice in extreme cases to use a large
bulletin-board or wall, and, having the information written on
scraps of paper, to arrange and re-arrange these scraps of
paper with thumb-tacks thereon, until the final order is settled.
Sometimes apparently complicated data turns out to be a
series of the combinations and permutations of simple elements.
Every step or event is then merely a combination of two or
three or more items of a descriptive nature. When these de-
scriptive component items are broken apart and listed indi-
vidually, it will usually be found that they are few in number
and can be grouped according to their nature into different
series, in such a way that one item from each series is present
in each event. Commonly, three of these component series
are sufficient to identify all the events. A production process,
MAKING IHE ODRVt OHABI
(Subject)
(Operation)
(Operator)
1
Data, Sources
Securing of
Chief
2
Computing
Instructions for
n
5
Execution of
Clerk
4
Checking of
n
5
Chart, Data
Inspection of
Chief
6
Field
Choice of
n
7
Data
Entering of
Typist
8
Checking of
Clerk
9
Scale
Choice of
Draftsman
10 ;
Curve
Plotting of
n
11
Checking of
Clerk
12
Scale
Entering of
Typist
13
Chart
Inspection of
Chief
14
Title
Choice of
1 **
15
Entering of ^
Typist
16
Chart
0. K,
Chief
Fig. 18 . A Tabulation of Simple Route-chart Data.
for example, may be made up of operator, object, and opera-
tion C'Vho,” ‘^Vhat'^ and ‘‘how”). Time (“when”) may also
be actually recorded. A distribution process may be made up
of combinations of place, person and proportion (“where,”
“by whom” and “how much”). The number and nature of
these component series will vary widely in different processes,
ROUTE-CHARTS
but the above are fair samples. Needless to say, where the
data can be analysed in this way, it will simplify the work of
compilation and assure the completeness of the data to list the
events, together with their component details, in parallel
columns, a column for each series or type of detail.^
A still more condensed type of work-sheet can be prepared
for complicated data, in which only two types or series of
descriptive items are present. This consists of a diagram in
which each series is listed fully and once for all, along an axis
MAKING TEE CURVE CHART
or edge of the paper. Thus on paper with columnar rulings,
list the descriptive items of one type or series (e.g. materials)
down the left-hand edge of the paper, and those of the other
type or series (e.g. departments) across the tops of the columns.
Then along the line of each first-series item mark with a cross
or circle the columns of those second-series items with which
it is combined to make an event or step (e.g. operation). It
makes little difference which series is put along either edge.
^ The student will be reminded of the keys and data-sheets used for maps and
diagrams, in which the two sets of values, or measurements along the axes, were listed
in column form, and will compare the descriptive detail series to the numerical value
series.
24
CHARTS AM) GRAPHS
If one series is longer than the other, it should be at the side,
but if both are of the same length, have the more important
series — the series by which events are to be grouped—listed at
the side edge of the paper. If this be reversed and the more
important series placed across the top, the crosses or circles
would of course be entered up and down the columns instead
of across them. The sequence of events can be shown by
small numerals in the circles, qr by connecting lines with
arrowheads, or best of all, by both. These connecting lines
MAKIHQ CURVE CHART
Fig. 20. A Very Condensed Work-sheet.
showing sequence would run horizontally if the side-edge
items are more important, vertically if the top ones are more
important. In practise either arrangement is satisfactory and
it is usual to leave the longer series listed down the page,
because more items can be written down a page than across it,
even when the column-headings are entered on edge.
The phenomenon of motion involves two inseparable ele-
ments, space and time, in either of w'hich the motion may be
ROUTE-^CHARTS
25
measured. It is ordinarily simpler to prepare first an analysis
of movement through space. This will include changes of
location, condition or operations. It is, in fact, simply a
measuring of events by arbitrary differences in their nature,
instead of a measurement by differences in point of time.
When it has been completed, it may also be desirable to
measure the movements chronologically and to co-ordinate
them upon the chart so as to show graphically the motion
through time as well as spa.ce. Assuming that the data for
the chart have been decided upon, we shall proceed to its
graphic presentation, beginning with the simpler route-chart,
showing only change of place or condition. For convenience,
this ma}^ be called the ^‘procedure-chart.
The simplest “procedure-^hart'" is a straight line or row of
“boxes” with the steps or events inscribed and with arrows
THL {.iAlN STAGES IiM CDnVE CHART MAKING
Fig. 21. The Simplest Procedure-chart.
along the connection-lines indicating the direction of move-
ment. It is important that the arrangement of the steps be
in a uniform direction across the paper. They can be arranged
horizontally from left to right, or vertically from top to bottom
(or even, in special cases from bottom to top). The same
considerations will determine this direction as were noticed in
the arrangement of the classification-chart; namely, the letter-
ing of the boxes generally gives them greater breadth than
height, and if there are but a few boxes, they can be placed
side by side. However, if there are many, they can be packed
closer one above the other.
The procedure-chart is in many respects similar to the
classification-chart, its main differences being that it need not
branch or split up at each new step, and that the connecting
lines between boxes indicate a path-way or line of motion.
For more complicated data in which the processes branch out
and split up, the similarity between the two charts will be very
great. The use of different styles or shapes of boxes now
becomes more advantageous, as the various steps may be
totally dissimilar, and by adopting certain shapes for each
26
CHARTS AND GRAPHS
1600 1810 1820 1830 1840 1860 18t,0 1870
1890 1500
type of step (e.g. operation) or for each, type of descriptive
detail (e.g. departments, persons, objects, functions, etc.),
these distinctions between steps or events are clearly brought
ROUTE-CHARTS
greatly to the value of the chart by picturing the various
CHARTS AND GRAPHS
:8
HhmiHng fhr aubjecls ircatcd in an article by Walter N. Palakcrv, in Engineerin'! ManagrmenC
by permission.
Fig. 24. A Graphic Outline of Thought.
stages realistically. At other times the chart is so simple that
the boxes can be omitted entirely.
A uniform direction of movement across the chart is im-
portant, because it automatically suggests to the reader the
sequence of events. It would better be described as a uniform
drift; motion at right angles to this drift, necessary at branch-
ings of the process, being immaterial. There are occasions
when, on account of the data, it is necessary to draw a line
backward, as is the case when seconds or by-products return
to an earlier stage for re-treatment, but these are legitimate
representations, suggesting actual backward steps of the
process. At other times it is necessary to choose between
backward directions of lines and repetitions of boxes; as a
rule the latter is the lesser of the two evils, but if the former is
decided on, the backward motion should be strongly indicated
by arrows, and the connecting-lines should leave boxes and
KOUTE-CHARTS 29
enter boxes at the points they would naturally leave and enter
if the boxes were in proper sequence.
Embellishments, artistic and otheiwise, are often met with.
Impartial study will usually show that nothing has been gained
by them, and that the message of the chart would have been
'f : '•
.V?
my.
yy/kA/R.
.vf- .
INDIAN REFINING COMPANV
SACES DEPARTMENT
DISTRICT AND STATION
l*erniii)hion oi Mr Richard WebUer
Fig. 25. A Popular Presentation.
I'he oiiginal of this chart, prepared in colors, is provided with a key explaining
the various channels through which influence is brought to bear upon consumers
by the sales department.
more strongly conveyed without them. The occasional ex-
ceptions to this rule are special cases of data in which the
subject-matter itself suggests the modified form as particularly
appropriate. In the “boiler-chart” the source of supply is
shown as a large tank or boiler, and the goods are shown as
flowing through pipe-lines to smaller tanks, cylinders, engines
and outlets, each representing a particular type of comparable
stage in the process. Here the reader’s imagination is fired
by the implied simile of a familiar mechanical process, and if
jO
CHARTS AND GRAPHS
the simile be a good one, he is likely to examine it closely and
so visualize the process clearly. Sometimes a row of tanks or
Permission of Mr, Malcolm C, Rcrrty,
Figr- 26. An Excellent Pictorial Route-Chart.
This shows the flow of supplies in the American Expeditionary Force,
vats will represent successive cost-burdens well, the overflow
(profit or- balance) from each flowing into and feeding the
ROUTE-CHARTS
31
Permission of Mr* Malcolm C, Rorty,
Fig. 27. The Analogy of Vats, Tanks or Reservoirs.
next. There is no limit to the possible variety of such repre-
sentations, but in the main the chart-maker will find the widest
play of his imagination called for in the making of legitimate
CHJRTS AND GRAPHS
procedure-charts, and will do well to avoid the excessive and
wasteful effort needed for the artistry of more sensational
products.
From Charts and Their Place in MananementP hy Frank />\ Gilhreth and X,. M. Gdbreth
‘ Mechmneal Xin^imering," Jan. 1922.
Fig. 2S. A Simplified Gilhreth Process-Chart.
Showing operations by conventional symbols and materials by pictorial drawings. '
The second type of route-chart differs from the first or
procedure-chart in that time is an important element in the
data and a feature of the chart. It may be called the “time-
chart,” It is easily made by arranging a time-scale along the
ROUrE-ClURTS
axis or direction of movement of the procedure-chart, and
adjusting the various boxes or entries of events so that their
positions coincide with the ordinates or abscissae of their par-
ticular points of time. The scale of time should be marked
along the edge of the chart (in large charts along both edges)
T-a aWT CHAPt
In practice this chart would bear dates for the days or weeks or other time intervals
at the top of each column across the page.
and straight lines in a faint color should be ruled across the
chart from the main divisions on this scale. A glance across
the chart will then show, by means of these faint lines, how
many time-intervals elapse between steps or events and a
study of the scale will give a more exact estimate when desired.
In time-charts, needless to say, the uniform direction of move-
ment or drift is essential.
The time-chart may be reduced to a form similar to that
of the work.sheet already described for procedure-charts.
With the time ruled off on one axis of the diagram, the items
to be followed by the chart are listed on the perpendicular
axis and the action of each item indicated by crosses, checks,
or solid shadings along the line of this item under or opposite
to the right moment of time. By various kinds or colors of
shadings, or by words alongside the shadings, the nature of
the event happening to the item can be indicated. In prin-
ciple, it is better to place time on the horizontal axis, so that a
standard time-scale can be used and long charts folded in
34
CHARTS AND GRAPHS
Curves -- 1./20 to 2/5/17
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Fig. 30. A Time Record, But Not a Time-Chart*
This is inserted for comparison with the previous illustration, as the dates form
the body of this table instead of the column headings, and the operations form
the column headings instead of the body.
sideways to reduce them to the same size records or files. If
very finely ruled paper be used, a large amount of detail can
be crowded into this sheet and if a number of similar processes
are to be compared, a standard arrangement of the items on
it will make quick comparison easy. This simplified form of
time-chart has little to recommend it from a graphic point of
view, but it will be found extremely convenient and sometimes
indispensable as a record, and is always useful as a work-sheet
in preparing a more graphic time-chart.
An example of this simplified time-chart is the Gantt
chart method. All charts in the Gantt system employ uni-
form vertical rulings, marking olF “time” on the horizontal
axis. The particular markings are always adjusted to the
individual business, so that the spaces between vertical rulings
may indicate hours, days, weeks, or months as is desired. And
the columns between these vertical lines, after their adjustment
to the periodicity of the particular business, resemble columnar
accounting sheets. At the left-hand edge of the paper, in a
very wide preliminary column, the machines, departments.
stock on Hand. January 1^+1981, Compared with One Year% Sales (AnAveraye of Five Previous Years)
ROUTE-CHARTS
materials, or other form of equipment are listed. And along
the lines of each of these items, under the proper “time” (that
is, in the column of the proper time unit) the process or opera-
tion is noted by a line commencing at the time of the beginning
3 ^
CIURTS JKD GRJPIIS
of the operation and ending at the time of its finishing. This
line takes the place of the crosses or circles above described
and has the advantage of showing by its length the length of
time required for the operation. The chart is used not only
for laying out work in advance in the planning department
and in the individual departments, but is also used to record
actual performance when the records of the same are being
kept. Wide lines are used in the place of narrow ones to
record actual performance.
c//A/rr FOR a hoi/r tc/rrs w/tr £4 rorrs off after a days, /r cd/ytyrdods
ORFRATfOR /RODS7R/ES
Permission of the Bureau of Labor Statistics.
Fig. 32 . A Simpie Time-Chart.
Another notable example of this type of chart is the Gilbreth
“process-chart,” used by Mr. Gilbreth to analyse methods of
work in his well-known micro-motion studies. On paper ruled
in tenths of inches, each fine line of which represents one-
thousandth of a minute in time, the length of time to perform
each operation or element of an operation is recorded by a
heavy line, beginning at the commencement of the operation
and showing by its length the number of thousandths of a
minute required to complete the operation or element of an
operation. Mr. Gilbreth ordinarily measures time vertically
down the page, a detail in which the present writer believes
him to be ill-advised. Across the page his vertical rulings
mark off the different parts of the human body whose motions
he is studying. There are about a hundred of these parts
listed at the top of the page, each over one narrow column or
space between vertical lines. Obviously the lines indicating
ROUTE-CHARTS
31
operations, extend vertically down the page under the parts
of the body active in the operation. Different colors of the lines
indicate different elements of the operation and different widths
of the lines indicate degrees of activity engaged in the operation.
Where the time runs in natural I'ecurrent cycles, such as
days, weeks, months, or years, and items re-appear at identical
points in each cycle, a circular form of time-chart is often
Fig. 33. A Weekly Clock-chart.
The weekly cycle of sales m a department store.
desirable. It has the advantage of being endless without
actually showing more than a single cycle and naturally
suggests to the reader the recurrent nature of the process.
In such a chart the time-scale runs around the edge of the outer
circle and the time-interval lines, equivalent to ordinates,
appear as radii from the center of the chart. The items or
events are inserted in boxes in their proper positions along
concentric circles, those near the center having of course less
room on the chart. Care should be taken to place the larger
items, or events, requiring more descriptive labelling, toward
the outside, if possible, in order that they may not be too
crowded. Such circular time-chartsarecalled ^^clock-charts"' but
they are not necessarily marked off like a clock; in one complete
revolution they will show twenty-four hours* for the day, or
Fig, 34. An Annual Clock-^hart.
The annual cycle of sales in a department store.
seven days for the week, thirty-one days for the month and twelve
months for the year, according to the time-cycle chosen.
Route-charts, like classification-charts, offer great freedom
to the ingenuity of the author. No set rules can be laid down
for their construction, though the general principles above
outlined will be found always safe and helpful. If the author
of the chart desires to modify it, he will break no iron-bound
canons, though he will probably have to do a great deal of
experimental work before he has a satisfactory product. The
one really final criterion by which his product will be judged
will be, as in all charts, how clearly, forcibly, and truly does
his chart tell his story. If he can pass this test better with a
novel form of chart than with the typical and sound forms
which have been described, he will have really invented a new
statistical instrument and his product will be a contribution
to the science, but the man with limited time will be well
advised to follow and remain within the fundamental principles
here outlined.
Chapter IV
COMPOSITE CHARTS
More fascinating than any one of the fundamental chart-
types already described are the results developed by combina-
tions of two or more of these types simultaneously. The simple
types are three in number, adapted to showing between
objects a space-relation (maps and diagrams), a topical rela-
tion (classification-charts) or a relation in motion (route-
charts). Any two of these relations may be shown simul-
taneously by combining the principles of their chart-forms.
It is only necessary to construct first one type of chart and
then with this as a basis or ground-work, superimpose upon
it the construction of another type, in such a way that while
each retains its own significance, the two harmonize in details.
Very often the two will be so closely interwoven that they
seem to be inseparable and indistinguishable, but the student
will always find that under close analysis they readily break
down into two or more separate and distinct charts belonging
to the essential types which have been described. He will
also find that this process of breaking down a composite chart
invariably clarifies his understanding of its subject-matter, and
vice versa, that the more obviously the component charts are
distinguished in the composite product, the more clearly its
subject-matter will be understood by its readers. If the chart
is composite, it is important that the maker should recognize
its nature, and it is important that the chart itself should show
on its face that it is composite.
If you will take a map of the country through which you
have travelled and with a heavy black pencil draw a line along
the routes you have passed over, with circles or boxes about
the names of places where you have stopped, you will have a
simple form of a superimposition. Had you marked upon
tracing paper over the map, instead of marking directly upon
the map, you would be able to lift your second chart bodily
39
40
CHARTS AND GRAPHS
Courtesy of Rand McNally Cs* Co,
Fig. 35. Route Map.
ofF of your base chart, and would see that you really super-
imposed a route-chart, describing motion, upon a map or chart
of space-relations. The result is a composite chart illustrating
motion through space.
Maps and diagrams are often used as a basis for charts
showing motion. In fact there has recently been developed
an elaborate technique of what are known as “pin-maps.”
These are particularly in vogue among sales-managers, who
have to route a number of salesmen about the country and
wish them to cover the most ground in the least time and with
the lowest possible travelling expense. Maps for this purpose
are mounted, and the markings upon them are made in the
form of conspicuous colored tacks and other devices driven
into or fastened onto the surface of the maps. In this form,
the same map may serve for many temporary superimpositions
and the latter can be readily altered at will, without the labor
of complete re-drawing, merely by removing or shifting the
adhesive markings. The labor-saving value of pin-maps is
so great for all kinds' of continual routing work that they are
marketed in excellent form by various commercial firms, includ-
ing nearly all map-making companies.
COMPOSITE CHARTS
41
Courtesy of Rand McNally Co.
Fig, 36. Pin Map.
The mounting of maps or diagrams to accommodate pins
or map-tacks should be closely examined. Ordinarily the
maps are mounted directly on wood, either to be framed and
hung on the wall, or they are already fitted into flat drawers
of special cabinets holding a large number of such drawers in
horizontal positions. The wood-mounting is, however, a poor
investment. Pins cannot easily be forced far enough into the
wood to be secure, nor easily removed if driven deep, and
sooner or later, as the wood shrinks under the punctured
paper, individual pins will drop out, and cannot be replaced
without complete rechecking of all data, the map meanwhile
becoming inaccurate and unreliable.
The ideal mount for a map, and in the long run the cheapest,
is a construction of cork and corrugated paper-board. The
map should be mounted directly upon a piece of cork linoleum,
with a non-wrinkling adhesive called rubber cement rather
than with paste or glue. The cork should then be backed
up with two or three layers of corrugated wrapping board or
paper, laid in alternate directions to prevent bending. In a
4-2
CHARTS AND GRAPHS
mount of this sort, the pins can be easily pushed into the map
to their heads; the cork grips them and prevents their falling
out, and the corrugated paper keeps their points from sticking
out underneath. Maps can be mounted in this way at home
or in the office, and in some instances can be procured directly
from the manufacturers of the maps.
Map tacks and other marking devices to attach to the
mounted map can be obtained in great variety, suitable for
showing a number of distinct markings and meanings at the
same time. The tacks are small steel pins with large round
or flat heads of cloth, celluloid, or, best of all, glass, conspicu-
ously marked, in different colors and sizes. When they are
inserted in the map to indicate, for example, towns on a sales-
man’s route, the various colors can be used to indicate diftei'ent
salesmen, the various sizes can show the length of the sales-
man’s visit, and the various markings on the tacks can tell
the extent of the company’s business there. Similarly, colored
string can be stretched between tacks to show the sequence
in which they are visited, while small celluloid rings of the
same color can be slipped over the tacks to show present loca-
tion or progress along the route. Ring-pins into which cards
can be fastened, are made in various shapes to hold cards at
different angles to the map, flat-headed pins on which labels
can be pasted, rough-ground celluloided pins on which pencil-
markings can be made or erased, and a wide variety of other
appliances are furnished for ingenious uses with mounted maps.
It is, however, no longer necessary to have mountings and
attached devices to make temporary and easily altered mark-
ings upon maps. Such arrangements take up too much space
and require special filing or housing equipment if used exten-
sively. Instead, it may be desired to use flat maps in book
or sheet form, which can be mpre easily carried about. For
this purpose a celluloid-coated map is made, the surface of
which is protected by a thin adhesive layer of pliable trans-
parent celluloid.i On this surface pen, crayon, or ink marks
can be made without difficulty and removed without injury to
the map, while gummed paper stars, dots, and other signals
in various colors can be attached and removed likewise.
_ ‘ The celluloided map is really little more than a map which has been surfaced
with a thin layer of liquid shellac or varnish. The liquid can be secured from dealers
in artisp materials and can easily be applied to any chart or map with a fine blow-
spray; it forms a protection against soiling, as the surface can always be cleaned
without removing the marks under the coating.
COMPOSITE CHARTS
43
The benefits of mounted maps suggest that floor-plans,
and other diagrams could be likewise profitably mounted and
used for pins and strings, but as a rule this is not yet a general
practice. The pathway of goods, papers, or functions about a
plant is generally marked upon photostats or blue-prints or
other reproductions of one original floor-map, in various
colored inks. The lines upon such diagrams should have
frequent arrow-heads to indicate the direction of movement,
as this direction is no longer shown by position on page as in
the simple route-chart. Different colors or kinds of lines can
be used to differentiate the pathways of various articles, but
when a large number of such articles are to be individually
followed, it is better to use a number of copies of the original
base-map, one for each article or group of articles, so that the
lines will not be too confusing.
An elaboration of this method has been described to the
writer by Dr. C. W. Gerstenberg, who tells of ‘"a factory where
a bird’s-eye view of each machine has been drawn to scale and
fastened with brass paper-fasteners to a piece of cardboard cut
to scale to represent the amount of floor-space needed for each
machine. The color of the card-board indicates the nature
of the machine; thus planers are on red card-board, drills on
blue, and so on. The mounted machine-diagrams are then
placed on a floor-plan of the factory and fastened into proper
position (they can be shifted if the machines are shifted), and
the routing of work is shown by ribbons slipped under the
machines and stretched from machine to machine in the order
of work. The flexibility of this ribbon idea commends
itself.’^
It may be added that if several types of work were routed
over this floor-plan, it might be advisable to use diflFerent
colored ribbons to distinguish them. Furthermore, some idea
of the volume of traffic or work along each route might be
given by using ribbons of various widths, narrow ones for
small or occasional vrork and wide ones for heavy traffic. The
student will notice that, in accordance with the principles set
forth below, the colors of machine-mounts should be in pale
tints and those of work-routing ribbons in brilliant tints, to
emphasize the route-chart over the floor-plan.
When several floors are to be shown on the same map, that
is, when the diagram must show in one plane a number of
surfaces which are really not side by side, but one above the
44
CHARTS AND GRAPHS
other, the co-ordinate or cross-ruled chart-paper can be dis-
carded and a special form of paper, with “isometric ’’rulings,
used in its place. This paper projects without any perspective,
three dimensions in space, and following the general plan of
charting on co-ordinate paper, can easily be plotted to show
the various floors suspended one above the other. A full
description of the principles of the paper will be given later,
but the paper is noted here for its peculiar value in this type
of composite chart.
Such route-diagrams often furnish the most forceful way of
presenting the weaknesses of a given arrangement of a factory.
Traffic congestion is apparent through the number of crossing
and confusing lines. A contemplated change which will result
in a more orderly march of goods about the factory will appear
upon such a chart with all its benefits made clear. A knowl-
edge of the chart-form will therefore be useful whenever a
revision or improvement of the lay-out of the plant is in view,
with the object of shortening transportation distances or of
installing “straight-line” processes.
COMPOSITE CHJRTS
45
Many other composite charts may be made besides those
showing motion through space. A classification-chart may be
Fig. 38- Routing on a Classification-chart.
Showing by its shadings, the departments through which the circulation of goods
and money takes place.
superimposed upon a map. A route-chart may be superim-
posed on a classification-chart. Two classification-charts may
be combined to show in a single chart both sets of logical
relations- Route-charts themselves may be combined to show
routings at different times or may be marked with distinctly
classifying features. There is no limit to the variety of ways
COMPOSITE CHARTS
47
in which the three principles of space, idea, and dynamic rela-
tions may be interwoven on a chart. Nor is the number of
possible strata or superimpositions necessarily limited to two.
We may occasionally need three or four distinct strata, and
indeed it is sometimes difficult to tell how many separate strata
have been superimposed to effect a single chart.
There are, how^ever, very definite rules which it is well to
follow in effecting the combinations. The first is that since
two distinct conceptions of relations are being expressed
through the same picture, the chart-maker should have clearly
in mind their comparative importance, and be prepared to
throw the emphasis in his construction upon the more im-
portant one. Failure to do this will result in a chart in which
the reader’s attention is distracted to the unimportant parts,
by which it becomes hopelessly confused in trying to follow
the important parts. The proper emphasis can be given by
using in the superimposed chart, heavier shading or stronger
lines which immediate^ distinguish it at a glance from the lines
and shadings of the base-chart. Where color can be used, it
should be used judiciously, after studying the effects of different
colors to see which gives the right degree of prominence.
The second rule is that since the superimposed chart gen-
erally carries the message, the base-chart is usually one with
which the reader is already supposed to be familiar. There-
fore, when the base-chart, as well as the composite, is unfam-
iliar to the reader, and particularly when a series of composite
charts is to be shown on copies of the same base-chart, it is
well to preface the composite charts with a single copy of the
base-chart, with which the reader may first become acquainted.
Then, when his attention is drawn to the composite, he will be
able to use it at once with a full understanding of its significance.
The conception of a basic or underlying pattern, as dis-
tinguished from a resulting superadded pattern, is important
and stays with us throughout the great majority of mathe-
matical charts which follow. The term ‘‘field” will be used
for this underlying pattern in these mathematical charts while
the upper stratum of curves, bars, and other markings will be
known as the “plotting”; and though the choice and construc-
tion of the “field” will sometimes be found more important and
often more difficult than that of the plotting, yet the field will
always be suppressed or submerged by lighter lines and colors,
to leave to the plotting its full significance.
PART 11.
AMOUNT-OF-CHANGE ANALYSIS
Chapter V
STATISTICS
Statistics is a word which has had an unfortunate history
throughout its brief existence. To begin with, its pedigree is
poor, for it is derived from old Latin words having nothing to
do with its present meaning. When it was created a century
or two ago, it meant ^^matters of State/^ or in diet ion ary-ese,
‘‘matters pertaining to the State.’’ It was then used for
those compilations of population, finance, and military strength
which rulers liked to have made about their various States.
But it has travelled far from that meaning, until today in its
proper sense it means any collection of figures and precise
numerical information.
The word was rapidly debased, until in common parlance a
statistician was one who carried at the tip of his tongue a large
assortment of appallingly uninteresting figures on widely
irrelevant topics, which he seemed to have memorized from
the encyclopedia with the ^le ^urjpose^of bo^^^^ Now
figures are not in themselves necessarily dry and dull — in fact
the figures of your bank-account may be very engrossing to
you. But figures on uninteresting subjects are a sure cure for
insomnia, to all of us. And it goes without saying that if the
figures are not of consequence, the chart of these figures will
deserve equally little attention. The point is that a chart is
as weak as its own data, and a chart-maker must carefully
weigh and consider his data before permitting himself the
pleasure of illustrating them with a chart.
But a worse charge than mere boredom is often levelled at
statistics. It has crystallized into a familiar saying, “figures
don’t lie, but liars figure.” Mark Twain went so far as to
remark at one time that there were three kinds of lies — namely,
lies, damned lies, and statistics — ^wicked in the order of their
naming. In short, the statistician is sometimes looked upon
as one whose acquaintance with figures is so very intimate that
48
STATISTICS
49
he can readily take liberties with them, abuse them, present
them in a false light, and deceive the layman. In this view
he is little more than a common trickster, performing leger-
demain with numbers, his magical results to be idly wondered
at, but not to be trusted. And the moral thereof is clear,
that he who would work with figures must be very? very sure
that his figures are all correct, both in their computation, and
their connotation.
As a matter of fact, surprising as it may seem, we are all
in the same boat, a whole nation-full of statisticians, in great
or small degree. We all perform mathematical operations,
arrange and study figures and precise data. We do it in our
accounts, in our reports, our decisions, and sometimes in our
sleep. If this be statistics, we are all guilty. And the odium
which sometimes attaches to the word must be only superficial,
for it does not attach to the practice or the subject-matter for
which the word is a symbol. Surely we will not allow our-
selves to be daunted by so empty a thing as a symbol or word.
Let him who will, retain his shallow prejudice and throw this
book aside here and now; and let the rest of us purge ourselves
once and for all time of any lingering superstition against
‘‘statistics.^" Let us resolve never again to utter a peep
against the word or to be terrified by its use.
There is, it is true, a more precise use of the word “statis-
tics"" in which a high degree of proficiency in handling masses
of figures is presupposed. In this technical sense, the statis-
tician is one whose ability to digest, compress, or extract
significance from a multitude of related numerical data, is
highly developed. The science of doing this is called by the
queer name of “statistical methods."’ In point of technical
skill it stands somewhere between the science of accounting
and the science of higher mathematics. If you have time to
dip into it you will find it more interesting than either, because
its applications are more varied than the former and more
immediately practical than the latter.i But for the purpose
of chart-making or chart-reading, it is not necessary for you
^ The notable books on the theory of statistics are:
Bowley, A. L., Elements of Statistics.
Yule, G. Udney, An Introductory to the Theory of Statistics.
Shorter and more elementary texts are:
Kelley, T. L-, Statistical Method.
King, Willford L, Elements of Statistical Method.
Secrist, Horace, An Introduction to Statistical Methods.
50
CILIRTS JND GR.iPHS
to have more than the usual grammar-school equipment m
mathematics — or as much of it as you have not forgotten.
That the maker of charts must dabble in statistics is
obvious; he who would build a house must first examine its
foundations. But the chief requisite in examining your stat-
istics is common-sense. You do not need deep mathematical
skill wherewith to perform difficult mathematical acrobatics.
Common-sense alone, without the aid of calculus or higher
algebra, will enable most of us to understand a falling bank-
balance, for example. Every school-child knows the meaning
of a total, or an average, and all adult persons ought to,
without regard for sex, color, or religious persuasion. Apart
from such elementary understanding of the meaning of the
language, the essential thing in ordinary statistical work is
common-sense. If you have it, you can traffic with figures
safely, and will often recognize a condition without knowing
the technical name or symbol for it, but if you haven’t it, all
the mathematical skill in the world w'ill merely befuddle you.
Of course, for much work, or for continuous application to
statistics, additional mathematical ability is an unquestion-
able advantage. It is a good thing to know, for instance, that
there are several kinds of averages, each with a meaning all
its own. It is well to have a nodding acquaintance with dis-
persion, the word nodding here being used to denote familiarity,
not sleepiness. And if you can shake a correlation coefficient
by the hand it may help tremendously in a pinch. The man
who can write an equation for his profits or his factory condi-
tions, has the edge on the man who has to make lengthy tabu-
lations. But the subject of this book is no more “how to be a
mathematician” than “how to get common-sense,” and we
will therefore drop both problems. We will split fifty-fifty on
them, and in proceeding with charting of statistics, assume
that the reader is gifted with common-sense, but not versed
in higher mathematics. Where special need for certain mathe-
matically technical terms or ideas arises, we will stop on the
spot and explain these terms, but we will not build our mathe-
matical bridges until we come to them.
One caution, and one caution only, in this chapter we wish
to make so clear that it will remain with the reader throughout
the rest of the book. That caution is, do not be afraid to use
your common-sense. In this matter an ounce of fore-sight is
worth many pounds of hind-sight. The value of this advice
STATISTICS
51
will come to you through long and bitter experience, anyway,
for it takes exceptional patience to scrap the results of weeks
of research and start all over again at the beginning, just
because of a failure to use common-sense beforehand. The
most important time to bring your common-sense into play is
before you lay pen to paper to take down a single figure.
That is the time to ask yourself the all-important question,
‘^What do I want to know?’^ Unless you can answer that word
clearly, positively and concisely, you had better wait until you
can, before doing anything else.
^What do I want to know?’^ Write it down in black and
white. Below it write the answer. Stand off* and look at that
answer as if you were a total stranger, and try to see whether
it makes sense. When you finally have the question and its
answer so clear that a child can understand, you will be ready
to compile and investigate statistics to substitute for the
answer. And when your work is finished, and you have boiled
down immense quantities of figures and numbers to a simple
coherent statement, see if that statement really answers your
question. If it does, you have statistics which may be well
worth illustrating with charts. If it does not, you had better
forego the pleasure of making charts, for it will all probably
have to be written off as a waste of time.
You may think this all very simple and easy, but you will
find it sometimes immensely difficult, and errors extremely
costly. No rules can be laid down, for every case is a matter
for individual study and analysis. But the consequences of a
mis-step at this stage are grievous. And it is right here that
so many amateur statisticians make their first mis-step. They
become lost in the zest of hunting up and chasing down all the
available related information; they allow themselves to be
dragged off the scent of the fox by every jack-rabbit that
crosses the trail. Red herring is their meat and in a shorter
or longer time they bring home a mountain of ^^statistics,’’ all
of which is ‘‘interesting if true,” but does not bear directly on
the point.
When you find this happening to you, you will be able to
recognise it by the bewildered sensation in your solar plexus
the first time a friend drops in and asks in a heartless way,
“What’s the good of it?” And right there is a good time for
you to stop — it would have been still better earlier — and ask
yourself again, “What do I want to know?” Think back once
CHARTS AND GRAPHS
5 ^
more to your original position and what you set out to learn*
It is never too late to mend. If you find you are on the wrong
trackj bravely scrap the work you have done, though it hurts
cruelly to do so, and strike out again for your goal
Your goal is a certain piece of information. Statistics are
merely the road to that information. The information itself
will ultimately reduce to a comparatively simple statement,
and if there are any statistics left in that statement, wipe them
out by substituting illustrations or charts for them. The
graphic method, whether used in the office to facilitate research
work, or in the published report to facilitate understanding,
should be confined to information which has value. It is
therefore an obvious but extremely important rule not to
begin a graph until the statistics have been carefully examined
and their object or significance brought clearly into mind.
Chapter VI
WORK-SHEETS
What would you think of a factory manager who kept his
plant so cluttered up \\ith raw materials, partly finished
materials, by-products, and working machinery that his work-
men had to climb over each other, and his materials had to
be passed from operation to operation by long forward passes
skilfully negotiated over ceiling-high piles of obstructionS|f
Yet this is precisely what most of us do in the far more difficult
case when oitr materials, workmen, and implements, are all
intangible ideas, expressed by strokes of a pencil on paper.
The old copy-book admonition to write clearly and neatly,
may often save you from utter confusion and defeat, and will
always greatly speed your work. Straight-line computing
methods are as important as straight-line factory methods.^
Statistical data generally comes in the form of long columns
of figures. If it is not already in this shape, you should so
arrange it at once. It may be, for example, the reports of
your sales in the various States of the country. By listing
the States in a column, you can write beside each one the figure
of its sales, and your sales will then form a second column.
Perhaps you wish to reduce these to per capita sales. In that
case, beside the figure for the sales for each State, you can
enter the population for each State, thus forming a third
column of entries (a second column of figures). The per
capita sales, which are merely the ratios between the two
columns of figures, can then be entered still further to the
right, forming a third column of figures (a fourth column in
all).
Frequently the computing is carried through a great many
steps, each of which calls for one or more columns of figures.
^ For an excellent discussion of the principles of tabulation, in addition to the
works on statistical methods already referred to (page 49), see Edmund E. Day,
‘^Standardization of the Construction of Statistical Tables/’ American Statistical
Association Quarterly, March, 1920, p» 59.
53
54
CHJRT6' JRD CIUPHS
toaiaiBijnos op i£bm noHtrx n tes mnw
ipprcaimto Stook* in Clxlef CowrstrlM, D«c. 31, 1513.
(Souro#: 0. S. Stati#ticeX AbutraotJ
Dollars
population
$
per
cap
tOTAl,
10,127,084,000
1,529,379,000
6*62
China
51,558,000
356,042,000
,09
India
176,634,000
516,166,000
,66
Ruacis.
411,600,000
178,905,000
2,50
Vaitad SUtaa
5,821,563,000
106,016,000
56,59
Oanaany
645/572,000
67,810,000
8.02
Japan
482,646,000
65,965,000
8,62
Auatria Hungary
64,734,000
62,368,000
1.26
Dutch Kaat Inciaa
49,202,000
47,956,000
1.03
Groat Britain
722,861,000
46,089,000
Ib.M
Pranoa
726,449,000
39,700,000
18.28
Italy
249,137,000
56,646,000
6.82
Braail
43,690,000
26,642,000
1.64
Turkey
21,274,000
*
Spain
658,861,000
20,60'J,000
52,H
Korea
23,889,000
16,913 ,0X1
1.41
Kexico
260,000,000
15,502,000
lb. 13
Egypt
59,376,000
12,566,000
3.13
Blasi
41,652,000
8,266,000
5.02
Canada
191,827,000
fi,075,000
23.7b
Argentina
521,869,000
8,0ub,000
59.90
Belglun
86,806,000
7,!j88,000
7.41
Ruaania
1,000
7,50K,000
e»
Neiherlaada
327,622,000
6,683,000
49,76
South Africa
53,543,000
6,465,OCO
6.16
AnitraXaaia j
246,422,000
6,976,000
41.24
Portxigal
49,254,000
S,9G8,0O0
8.28
Peru
32,691,000
6,800,000
6,63
Swedes
88,866,000
6,713,000 1
16.66
Coloobla
10,768,000
5,071,000 ;
2.12
Uorocco, fresoh
24,638,000
6,000,000
4.93
Serbia |
55,488,000
4,822,000
T.26
Ceylon
5,776,000
4,262,000
1,56
Svltaerlaad
121,283,000
3,880,000
51.26
Fomoaa
34,092,000
5,711,000
9.19
Chili
11,363,000
3,641,000
5,12
Finland
5,269,000
•
Denmark
52,649,000-
2,921,000
18,02
Joliria i
2,890,000
*
Taaesuela
21,646,000
2,616,000
7.69
Korwiy
44,911,000
2,609,000
1T.89
Haiti
860,000
2,600,000
•34
Qtaatemala i
2,119,000
*
Xaxiador 1
4,140,000
2,000,000
2.06
Dmgxtay
51,094,000
1,546,000
37,96
Salvador
4,598,000
1.268,000
5«46
Paraguay !
482,000
1,000,000
.48
Doninloan nepublle
800,000
725,000
I.IO
ntraita bettleoentf
17,265,000
714,000
24.18
Kioaragua j
704,0OCr
Honduraa
•
682,000
Coata Kioa !
2,112,000
451,000
4.90
tiDceBbourc 1
1,865,000
200,000
T.IT
Brltiah Honduras {
168,000
41,000
4.09
Fig, 40. A Simple Computing Sheet.
The author has, he regrets to say, actually carried one investi-
gation through so many consecutive steps that the resulting
columns of figures, when pasted as close together as possible,
side by side, without repeating any column, reached completely
around the walls of an ordinary room. There is generally no
excuse for as much work as this, but it is well to bear in mind
that every computing step may call for two or three columns
of figures, and that if you are going to carry your figures
through many steps, it will pay to have them in uniform
columns.
WORK^SHEETS'
55
Obviously the arrangement of States or items in the colutnns
should be standard throughout the work, so that any two
columns can be readily compared. This makes the order of
the items a matter for careful study. In listing the States of
the union, for example, you have your choice of three arrange-
ments. In the first place, you can list the States alphabetic-
ally, which makes it easy for a stranger to find any particular
State at once. Secondly, you can arrange them in the order
of their importance (as viewed from the particular stand-point
of your problem), which makes it easy for a stranger to focus
his attention at once on the important States. But neither
of these methods, though widely used, has anything more to
recommend it than a certain possible convenience to strangers.
For computing at least, the States should be arranged in a
logical order, and a logical order is generally one that brings
nearby States together, instead of scattering them about the
list. The reason for this is that you may find you want to
take off sub-totals (totals for East, North, South, and West)
to get group-figures for groups of States. And if the States
are already arranged or grouped together this is easily done.
In fact, it is always well to carry these group-totals, because in
checking through to locate errors they save much time.
Having decided to arrange the States logically by territorial
groups, you have next to decide how to group them. The
census has one grouping, dividing the country into six or
seven territories. But this grouping is not the best natural
economic one, that is, it does not conform to natural business
groupings. The Audit Bureau of Circulations has another,
which is, for general business conditions, perhaps the best.
But in most cases there will be individual factors which make
it desirable to adopt a special arrangement of one’s own.
Large sales organizations, for example, will already have set
up their own sales districts and branch house territories, and
in such cases it is best to make the grouping of States conform
as much as possible to these.
Of course, not all tabulations are tabulations of States by
State figures.' These figures illustrate, however, the principles
of tabulating. We may have to work with figures, year by
year through a series of years, or with month-by-month
figures, through one year or more. In these cases, where we
are working with divisions of time, the natural and proper
way is to place the earliest periods at the top of the list and the
56
CHARTS AND GRAPHS
u.s.
Audit
Bureau
Census
of Circulations
Red Cross
Ala'bama
NEW ENGLANIT
ErGLi:©
NEW ENGLAND
Arizom
Maine
I.faina
Maine
Artensaa
New Hampshire
New Hampshire
Massachusetts
California
Vermont
Vormont
Hampshire
Colorado
Massachusetts
lassachusottfl
Rhode Island
Connect lent
Rhode Island
Rhode Island
Vermont
Delaware
Florida
Comectlcut
Conneotiont
ATUNTIC
Georgia
MIDDLE ATLANTIC
NORTH ATLANTIC
Correct icut
Idaho
New York
Hew York
New Jersey
Illinois
Hew Jersey
New Jersey
New York
Indiana
Pennsylvania
Pennsylvania
loia
Delaware
PENN. -DELAWARE
Kansas
BAST NORTH CENTRAL
Maryland
Pennsylvania
Kentucky
Louisiana
Ohio
Indiana
District of Col.
Delaware
Maine
Illinois
SOUTH EASTERN
POTOMAC
Maryland
Michigan
Virginia
Pistr. of Col.
Massachusetts
Wisconsin
North Carolina
Maryland
Michigan
South Carolina
Virginia
Minnesota
WEST NORTH CENTRAL
Georgia
loot Virginia
Mississippi
Minnesota
Florida
Missouri
lown.
SOUTHERN
Montana
Missouri
SOUTH WESTERN
Florida
Nebraska
North Dakota
Kentuclcy
West Virginia
peorgia
Nevada
South Dakota
North Carolina
New Hampshire
Nebraska
Tennessee
South Carolina
New Jersey-
New Mexico
Kansas
Alabama
Mississippi
Tenneosee
North Carolina
SOUTH ATUNTIC
Louisiana;
LAKE
North Dakota
Delaware
Texas
Ind.ana
Ohio
Maryland
Oklahoma
Kentucky
Oklahoma
Oregon
District of Columbia
Virginia
Arkansas
Ohio
Pennsylvania
West Virginia
MIDDLE STATES
CENTRAL
Rhode Island
North Carolina
Ohio
Illinois
South Carolina
South Carolina
Indiana
Iowa
South Dakota
Georgia
Illinois
Michigan
Tennessee
Florida
Michigan
Nebraska
Texas
Wisconsin
Wisconsin
Utah
EAST SOUTH CENTRAL
Minnesota
Vermont
Kentucky
Iowa
GUIF
Virginia
Tennessee
Missouri
Alabama
Washington
Alabama
North Dakota
Louisiana
West Virginia
Wisconsin
Mississippi
South Dakota
Nebraska
Mississippi
Wyoming
WEST SOUTH CENTRAL
Arkansas
Kansas
HORTSERN
Minnesota
Louisiana
WBSTSEN STATES
Montana
Oklahoma
Montana
North Dakota
Texas
Wyoming
Colorado
South Dakota
MOUNTAIN
New Mexico
SOUTHWESTERN
Montana
Arizona
Arkansas
Idaho
Utah
Kansas
Wyoming
Nevada
Missouri
Colorado
Idaho
Oklahoma
New Mexico
Washington
Texas
Arixona
Oregon
Utah
Nevada
PACIFIC
Washington
Oregon
California
Oallfornla
MOUNTAIN
Colorado
New Mexico
Utah
Wyoming
NORTHWESTERN
Idaho
Oregon
Washington
PACIFIC
Arizona
California
Nevada
Fig. 41. Various Geographic Groupings of the States.
WORK^SHEETS
SI
latest at the bottom, with possibly space for quarterly, half-
yearly, or five-yearly totals, which ever may be desired. Still
other types of items may occur, such as in a list of the various
departments of a plant, the various salesmen of a selling
organization, or the various products, and so on.
The point is that whatever the items be with which we
work, they should be arranged carefully at the outset, so that
it will not be necessary to alter their arrangement later. They
should be placed in a column if possible, so that the computing
and tabulating can be made in parallel columns beside them.*
If the list is long, it should be broken up by blank spaces at
convenient intervals, or better still, the items should be
grouped together, for which sub-totals may be required, and
blank spaces should be inserted between the groups for these
sub-totals or part totals. But by all means, try to get the
whole list on a single page, even at the cost of pasting additional
sheets at the bottom, for the work will progress much faster
on one large sheet which is complete than on several small
sheets which are not complete.
If much work is going to be done, it pays — and paj^s well —
to have the printer rule up some sheets with the list of items
printed at the edge of the sheet and the lines ruled in where
they will be useful, horizontally from each item. It is a small
matter, but worth noting, that the lines should be regular
typewriter distance apart, so that should you wish to have
figures typed on these sheets the typist can work rapidly and
neatly. More important still, where adding machines are
(Tone)
SEW ENOLAND
H. Hampehire, Uass., k Conn.
MIDDLE ATLANTIC
10,800
N«v York A New Jersey
2,600.000
PennsylTanla
80DTH-EASTBRM
14,000,000
Maryland
524,000
Vireini*
429.000
Alabama
2,390,000
Tennessee
283,000
Kentucky
772,000
MOBTIl CSNTBAL
Ohio
8,650,000
Indiana A MichiEMi
2,940,000
Xllihoi*
3,280,000
mSSTERN
2owb« Missouri. Colo.. Mont.. A Ore.
465,000
PIO-IROS PRCtmCTIOH
• Unlttd State*
1920
F%. 42. An Incomplete State4ist Geographically Arranged.
58
CHARTS AND GRAPHS
used, is to get the lines spaced In the same way that the
adding machine prints the tape, so that figures do not need
to be copied from the tape, but the tape can be pasted right
on the sheet. In wide carriage machines, the tape can be
dispensed with, and the figures printed directly by the adding
machine on the sheets. These are devices to speed up the
work, which are trivial in themselves but very important in
their results, and, with a careful eye to your equipment, you
will soon hit upon the most useful forms for doing your com-
puting, standardise them, and call in the printer to prepare a
number of blanks.^
We have here considered carefully only the matter of the
items which flank the left hand edge of your work-sheet forms.
Technically, these items are called '‘the stubs,” indicating that
they are the labels attached to each horizontal line, or row of
figures in the columns or column of the table. The whole
sheet, filled with figures in orderly arrangement, is called a
“table” or tabulation. The vertical lines of figures are called
“columns” or sometimes “arrays,” while the horizontal lines
of figures, that is, the sets of figures beside each stub, are called
“rows” or lines. And at the top of each column of figures,
the label which describes the column in the same way that
the stubs describe the rows, is called a “heading” or “caption.”
A whole chapter could be written about captions, or column
headings. It is a fine art to make them at once clear and brief,
and to arrange them so logically that the thought moves easily
from one heading to another. In most cases of several columns,
two or more captions can be bracketed together by a third
common group-caption. In this way, the headings often take
on the form of miniature classification-charts. The best prac-
tice is to box in the headings carefully, so that they will be
clearly understood. In work sheets they should be arranged
^Tht* pQ^irlon of fhc total (or sum) in a tabulation is an important matter. There
are two possible positions: first, at the beginning or top of the table; second, at the
end or bottom of the table. The first is the statistical position; it is correct for a
published table, as it places the most important item, the total, or whole, first
before the reader’s eye. When parts are themselves further subdivided, their totals
should also be placed before their parts. The details, or parts, can be in every case
further indented than the totals. This practice should be adhered to in charts and
in published or recorded tables.
The second position, at the end of the table, is the accounting position. It is
correct for all cases in which the work of summing up the parts must be frequently
undertaken. Needless to say, it is essential for forms to be used in adding and listing
machines, and is advisable in general for work-sheets.
/FORK-SHEETS
59
iwrtmm n* ths OHit£D staiss
XXHiera.te percaata^e of eaoli olaas (Ijy aga, box, raooj of the population
for eaoh group of Btatee, 1920
(Source:- U. S* Ceueus)
YouUi
Adalt
[ Total population oTor 10 yoaxs' of age.
(Aged
16-21
years)
{6v0v 21 years)
Foreign
horn
■whit©
f
Male.
Fenale.
■white
Kogro
Total
TOTAL D. S.
8.S
7.0
7.8
2.0
13.1
22,9
6.0
Bsv Engluul
1.1
6.0
6.2
0.7
14.0
7.x
4.9
Middle Atlantlo
0.6
6.9
6.6
0.6
16*7
6.0
4.9
East North Central'
0.6
8*7
8*6
0.9
10.8
7.8
2,9
West North Central
0.6
2*6
2.6
0.9
6.4
10.6
2,0
South Atlantlo
•r.9
14*0
15.9
8,1
12.8
26.2
U.6
East South Central
T,6
16.7
16.2
6.4
9.1
27.9
42.7
West South Central
7.8
12.8
12.1
4.1
29.9
26.S
10.0
Mountain
8 .7
6*4
6.8
2.0
12.7
6.8
6.2
PaoiflB
1.2
8.3
3.0
0.4
6.6
4.6
2.7
Fig. 43. Classified Headings to the Columns.
in the order in which they will be computed, so that the work
of computing moves as much as possible to the right, always
preferably deriving each column from the immediately pre-
ceding columns. Where two or more columns, which are in
themselves the results of several columns, must be combined,
each can be left at the right-hand edge of separate computing
sheets and then by cutting off the remaining paper, or by fold-
ing it back, you can lay the sheets one over the other so that
the columns to be worked over will appear side by side.
One of the cardinal rules for computing tabulations — and
it applies only in lesser degree to final tabulations ready for
publication, presentation, or study — is that every column head
should include a number or letter identifying the column.
This rule has even been extended by some authorities to the
stubs as well. The advantage of the number or letter is that
it makes reference to the particular column very easy, either
in texts, notes, conversation, or formulae.
In addition to a symbol for the column, you should also
have in your column-head, a note explaining the source of the
figures in the column. This note can either be the name of
the authority from which the figures were copied, or it can be
the formula by which the figures were computed from other
columns. It will save you much trouble in correcting the
errors of computing clerks, and much time later on when you
come to refer to the sheets and do not remember the various
THE t^UKGEH GENERATION
Co
CIIJRTS JND GRAPHS
Column Symbols, Formulae, and Classified Stubs and Captions^
WORK^SHEETS
6i
steps clearly. For the clerks who are doing the computing,
this note is a standing bill of instructions. And the time will
SAVIKOS BASK STATISTICS
Stmber of saTingc banks and eavings bank dapositora; total, average, and pereapita
deposits; and ratios between banks^ depositors, and population as specified belov.
United States, 1820-1920
( Arrange
d from U. S.
Statistical Abstract
Per-
Pop.
Average
j Depositors per
Year
B&nlcs
Deposits
Depositora
depoait
ropuXciXjiOn
deposit
Pop.
Bank
bank
Number
Dollars
Number
Dollars
KuDber
Dollars
Percent
Number
Humber
Copy
Copy
Copy
B/C
Copy
B/E
100 C/E
C/A
E/A
A
B
C
D
E
P
0
H
J
1820
..
1,138,676
8,635
132
9,638,463
.12
.09
..
1828
16
2,637,082
16,931
160
11,150.000
.23
.16
1,130
744,000
1830
36
6,973,304
38,035
183
12,866,020
,63
.30
1,066
357,000
1835
52
10.613,726
60,068
172
14,710,000
.72
.41
1,155
283,000
1840
61
14,051,620
78,701
179
17,069,463
.82
.46
1,290
280,000
1846
70
24,606,677
148,206
169
19,970,000
1.23
.73
2,076
286,000
I860
108
43,431,130
261,354
173
23,191,876
1.87
1.08
2,326
216,000
1856
215
84,290,076
431,602
196
27,256,000
3.09
1.68
2,050
126,700
1660
278
149,277,604
693,870
206
31,443,321
4.76
2.20
2,495
115,100
1865
317
242,619,382
980,844
247
34,748,000
6.99
2.83
3,096
109,600
1870
617
649,874,368
1,630,846
337
38,658,371
14.26
4.24
3,260
74,500
1675
771
924,037,304
2,359,864
408
43,951,000
21.00
6,37
3,060
56,900
1880
629
619,106,973
2,335,682
366
’ 50,165,783
16.30
4.66
3,710
79,800
1885
646
1,096,172,147
3,071,495
366
66,148,000
19.60
5.47
4,760
j
87,000
1890
921
1,624,844,606
4,258,893
369
63,056,438
24.18
6.69
4,630
66,400
1895
1,017
1,810,597,025
4,875,519
372
69,579,668
26.04
7.01
4,790
68.400
1900
1,002
2,449.647,886
6,307,083
401
76.129,408
32.18
8.02
6,100
76,100
1605
1,237
3,261,236,119
7,696,229
424
84,219,378
38.80
9.12
6,220
68,100
1910
1,769
4,070,486,246
9,142,908
445
92,267,080
44.20
9.91
6,200
62,600
1916
2,159
4.997,706,013
11,285,755
443
99,342,625
60.40
11.35
6,220
46,000
1920
1,707
6,636,470,000
11,437,556
671
106,418,175
61,20
10.73
6,700
62,300
Fig. 45. Column Symbols and Computing Instructions.
come when you will map out your work in this way, merely
filling out stubs and the column-headings yourself, and leaving
the entire computing task to clerks.
It is also just a question of time before you will come to
look upon these tabulated figures as so many various descrip-
tions or phases of the original set of stubs. The column-
headings tell you the type of description or phase, but in the
end the stubs are the basis, and the figures are derived from
them. Mathematicians have a word “function” for such rela-
tions. Using that word, we would say that the tabulated
CHARTS AND GRAPHS
62
figures are functions of the stub-figures or items, meaning
that their values are derived therefrom. Glancing across the
various lines, you will see that all the figures on the same line
with a stub are but various and varying functions of that stub,
the symbol at the top of the column identifying the function
and the caption describing it.
Glancing down any column, you will see that the figures
change from item to item. They form a series of varying
values. Each column contains the various values of the func-
tion described by the caption head. Each column can be
looked upon as a series of figures which are the readings or
values of the item, described by the column head. And the
point is that while the stubs, and captions, were independently
arranged by yourself, the figures in each column are derived
from or attributed to them and so are dependent upon the
stubs and captions. In short, while the stub is an ^'independ-
ent variable,’’ the series of figures in the other columns are all
"dependent variables” with regard to the stub.
Sometimes the independent variable is called the
variable” and the dependent variable the "y-variable.” In
that sense the values of "y” will all depend upon the values or
meanings of We might go so far as to label the column
of stubs ".r” and the captions "y,” being various kinds of "y”
variables. Think of the first one as "y,” the second column
as "yi,” the third as "y2>’’ the fourth as "ya,” and so on, and
it will always be clear to you that the stub is the independent
or jjc-variable and the other columns are the dependent or
y-variables- A second table on an entirely different aspect of
the same items or stubs, could be called a ' Vvariable,” meaning
that while it was also a dependent, it was distinct from the
first set of dependents. As a matter of fact, this is all relative,
for at some time you may treat one of the columns of figures
as an independent variable and the stub as its dependent.
But it is useful to begin thinking of your tabulation as having
both independent and dependent Variables contained in it.
Chapter Vll
CO-ORDINATES
The better to demonstrate their simplicity, we have dis-
cussed in the very first chapter those few technical terms which
will be essential to the student in this elementary work. In
that chapter the analogy is drawn between the checker-board
arrangement of streets in certain American cities and the criss-
cross rulings of most chart paper. In this chapter this par-
ticular type of ruling will be carefully re-considered for the
benefit of those whose zeal in the subject has led them to skip
the first chapter, and a few other forms of ruling will also be
touched upon with their relations to this fundamental system.
The chapter will not be interesting reading, but it deserves
close study in order that the remainder of the subject may be
clearly understood.
Station yourself in an open field where you can, without
difficulty, move in any direction — even upward with the aid
of an aeroplane or downward with the aid of a good fast shovel.
At the point where you stand, drive a stake into the ground
and consider it your base of operations for all other points on,
above, or below the field, that is, the point from which you
can measure their distances. We will call it the ‘^point of
reference” or ^‘point of origin” and mark it ^^O,” which accord-
ing to your taste may either stand for the word ‘‘origin” or
for its zero distance from the origin.
Facing in any direction at this point walk forward a dis-
tance, let us say, of five steps, in a straight line. It is obvious
that you could walk forward indefinitely in this direction,
always reaching a greater distance from the starting-point “O,”
J j 1 j j 1
0 12 3 4 5
Fig. 46.
and that you could always measure this distance by stakes at
each foot-step numbered “1,” “2,” “3” and so on successively
63
t)4
CHARTS AND GRAPHS
as you pass them. You could then instantly locate any point
along your path by the number of the stake. Points midway
between full steps can be given fractional values. Such points
would be definitely and unmistakably identified if, in addition
to telling their distance from the origin, you also tell the direc-
tion in which you had walked when you left the stake to
measure them. Calling this direction you would specify
the points completely by calling them 3P and ^
so on.
Having walked forward, however, only five steps, you
would reach the point 5.’’ Suppose here you stop and begin
to retrace your steps, walking backwards. You would notice
that now instead of the distances from increasing, they
decrease as you walk backwards, until after five backward
steps you are again at the zero-point. But continue to walk
backward and you will have again the phenomenon of increas-
ing distances from this point as your steps increase. In other
words, along the same straight line, there are two sets of
distances mirroring each other at the zero-point. If you walk
backward ten steps all told, from the point come to
another spot which is also five steps from the origin and in the
same straight line. There must be some way to distinguish
the two and to distinguish all the other pairs of distances in
this straight line which we have called Suppose that for
this purpose every distance reached by walking forward from
the origin be called positive and every distance reached by
I 1 1 , j 1 1 \ H — 1 1
•5 .4 -8 -8 .1 0 *2 *8 *4
Fig. 47.
walking backward from the origin be called negative. Leaving
the first set of stakes as before marked ‘‘x, 1” (or “x, +1”),
“x, 2” (or “x, +2”) and so on, you can similarly mark the
stakes at your backward steps, from the origin, by numbering
them “x, -1,” “x, -2,” “x, -3” and so on. In this way you
will easily identify every point along the straight line, both
on one side of the origin (in the “x” direction) and on the
other side (in the “ -x” direction).
Here you have set up measurement in one direction, that
is, along one “dimension.” There is a technical name for this
“direction” or “dimension,” which it will pay you to learn,
namely “axis,” In fact, we can already speak of it as the
CO-^ORDINATES
(>S
^ or axis of measurements. But leaving aside
technical terms, your common-sense will tell you that you
have set up the only kind of measurement possible in one
dimension, namely “linear measurement.’' In this particular
case, your foot-step — one half of your pacing distance — is
your “unit of measurement.” Any other unit could have been
taken. You might have laid a certain stick down repeatedly
and marked off the number of times it could be laid end over
end. If you had a number of sticks of the same length you
could lay them end to end and leave them lying to form one
long pole or rod cut into equal parts. But whatever the
length of your unit of measurement may be, you will, by num-
bering the units successively in both positive and negative
directions, have graduated or “calibrated” that one long im-
aginary rod with a “scale” — the scale or calibrations being the
numbers or countings from the zero-point in. both directions.
It will occur to you, however, that while you have an excel-
lent system for locating points in that one line along which
this imaginary rod lies, that is, along which you walked, you
have no means of identifying points elsewhere in the field.
Suppose, therefore, you return to the “origin” with a short
actual rod which is long enough to reach from point “x, S” to
point “;r, ~*5” and lay it actually between those two points.
(You can extend this rod in your imagination indefinitely, in
both directions, but a rod of ten steps length is handier to
carry about than an indefinitely long one.) At every stake
mark the corresponding point on your rod, with the same
numbers, so that you can dispense with the stakes entirely,
and with the rod in this position you can still locate points
along the “;r” direction from the origin, either positively or
negatively, at once.
Now stand at the origin with the positive markings on the
rod to your right and its negative markings on your left hand,
and you will find yourself facing in a direction at right angles
to your first line of walk. Strike out in this new direction.
As 3 rou walk you can again keep track of the distance by count-
ing foot-steps, and you will again in this new direction find
both positive and negative values which mirror each other.
You can find positive values by walking forward and negative
ones by walking backward from the origin. This new direc-
tion, lying at right-angles to the original direction, yoii
can call the “y” direction, and specify points along it as “y, 1,”
66
CIL-iKTS JND GIUPHS
“y, 2,” “y, 3” and “y, -1”, -2,” -3/’ and so on.
And so you will have set up a second scale at right angles to
your first or “%-scale,” by means of which you can identify
points along the “y” direction from your origin. The unit of
measurement may or may not be the same as the unit used in
the first measurement, but of course whatever unit you adopt
in the “y-scale” must be adhered to as closely therein as was
the “x;-unit” along the “;c-axis.” Being in a new direction,
the distances have no relation to those in the old direction,
but it is obvious that along either direction the units therein
must be uniform. And since we called the first direction the
“x:-axis,” we may call this new one, at right angles to it, the
"y-axis.”
Now if you will pick up the rod, lying in the “x-direction”
and carry it with you as you walk in the “y-direction,” without
swinging it or moving one end faster than the other, but taking
care to hold it rigidly as you walk, you will see that every
point along the rod, marking distances along the “x-axis”
describes a straight line parallel to the “y-axis” in which you
walk. Here you will have a means for identifying any point
on the field. You need but to carry the rod or x-axis out
along the y-axis until the rod crosses the point you wish to
identify. Then note the point on the rod with which it co-
incides, such as “x, 3,” and the point on the y-axis to which
CO-ORDINATES
67
you carried the rod, such as 6/^ You will define the point
as being y, 6/’ and you will search in vain for any other
Fig. 49.
point which can be similarly described. Three other points
you will find which have the same numbers, but not the same
signs; these points will be 3; y,-6, ‘‘.r, -3; y, 6'’ and
—3; y, —6.’’ You will find that your two axes cut the
field into four quarters, in each of which all points have the
same combination of signs.
If you do not wish to carry the rod back and forth along
the y-axis, or a similarly marked rod in the y-axis back and
forth along the ^-axis, you can lay a series of rods parallel to
each other and perpendicular to the ;c:-axis, crossing that axis
at each point marked off on it, and another set of perpendiculars
along the y-axis so that your entire field is crossed by these
measuring rods in two directions. Thereafter you can locate
any point on the field by walking out either axis and then
turning at the right distance and following the perpendicular
there, parallel to the other axis.i
You now have at your disposal a means for identifying any
point or any number of points upon a given field, or in precise
^ The co-ordinate axes need not be at right angles to each other, they may be drawn
at any other angles desired; in the latter case, they may be thought of as really at
right angles, but seen from a side rather than a direct view.
68
CHARTS AND GRAPHS
language, in a plane surface. A plane has, as geometricians
say, two dimensions, commonly called length and breadth.
-4 -8 -S .1 0 1 3 8 4 e
Fig. 50. Field with Equal Scales.
For these you have laid down two straight lines, or axes, at
right angles to each other. One of these you have called the
“x-axis,” the other the “y-axis.” Both have been calibrated
or measured off and scales attached. And with this simple
mechanism you can take measurements of various objects in
your field and record locations so precisely that others can,
by your records, be led to the very same objects, or can dis-
cover without possibility of doubt, the exact spots upon which
the objects had been placed when you measured them. In
4.IS w4 -8 -a -1 0 1 2 8 4 fit
Fig. 5U Scales pf A?ces Unequal,
CO-^ORDINATES 69
short you have the means for identifying any point upon a
plane surface.
It is a fortunate coincidence that the paper upon which we
ordinarily write is flat and its surface can be considered a plane
surface. For this enables us to apply a reverse English to our
measuring device, and use it in making precise illustrations of
such fields as we have measured. Now instead of being given
a field with objects and being required to set up the measuring
device in order to ascertain the location of the objects, we will
be given paper already ruled off with the measuring device,
and will be required to place thereon indications of the objects.
The paper ruled off in this way is called “co-ordinate paper”
for reasons which will presently appear. Its rulings represent
a series of parallel lines laid crosswise over another series of
parallels, and we are at liberty to select any two intersecting
lines for our axes and mark off our scales or measurements
along these lines as large or as small as may suit us. And you
will find that more than half the charts you ordinarily encounter
will be constructed in this way. The horizontal lines are called
abscissae, the vertical ones ordinates.^ Taken together, these
co-ordinate rulings form what is, in chart-making, technically
called the “field” of the chart, being the background upon
which the distinctive portion of the chart is superimposed.
And as will be remembered, from the chapter on superimposi-
tions, the basic portion or field should be as unobtrusive as
possible, that the important features may receive more atten-
tion. The field should always be drawn lightly, with thin
lines, and with no more co-ordinates ruled in than are necessary
to afford the chart-reader ease of comparison. If possible,
the field should be in green or grey ink, as this further sub-
merges it.
The origin or zero point we have so far taken within the
field, so that the field is cut into four quadrants in which the
2 In consequence of that quaint genius for unnecessary trouble sometimes exhib-
ited, a very serious discussion has occasionally arisen as to whether the abscissae
should be ruled upon the paper with their upper or lower edges upon the exact positions
which the lines signify. The idea of those who favor the upper edge appears to be
that the abscissae are shelves upon or above which plotted points rest, when permitted
to do so. The idea of those who favor the lower edge is not so cogent. Curiously
enough, similar debate has never raged around the position of the ordinates. As a
matter of fact, few charts are so precisely drawn or so finely adjusted that the thick-
ness of the ruled line is material, and in every case, the obvious place for all ruled
lines is a centered position, that is, one in which their two edges are equidistant from
the precise desired positions of the imaginary lines they represent.
70
CHARTS AND GRAPHS
ij. 52, Origin of Chart near One Edge of Field.
values of points mirror and repeat themselves with dilFerent
plus and minus signs. But in practice a large proportion of
our measurements or data present no negative values^ at least
Fig. 53. Origin in Comer of Field,
along the x-dimensbn. The right-hand upper quadrant, or at
most the two right-hand quadrants are then sufficient, and we
can omit the remaining quadrants entirely. As a result, the
ordinary chart on these co-ordinate rulings shows the value of
zero or. its origin-point along its left hand edge and generally
in the lower left-hand comer. The method is still the same,
but a portion of the field is merely being omitted, because
useless. In fact, we can even go further and begin the chart
Fig# 54# Origin Not Shown in Chart.
out to the right of the origin, omitting the origin itself when
that also is not to be used.
This system of parallel lines to two axes, which are them-
selves at right angles to each other, belongs to what is called the
Fig. 55. Field with Co-ordinates Not Perpendicular.
Cartesian system of co-ordinates. That system need not stop
with plane two-dimensional surfaces but can be extended to
cover three-dimensional space, by the very simple expedient
of perpendiculars erected at the intersections of the parallels.
You may think of this as lifting your entire net-work of rods
up over your head as you stand on the ‘^origin” in your field,
or forcing it down into the earth below you; calling the vertical
line through the ‘ Vaxis’' and measuring upward distances
positively and downward distances negatively. But because
three dimensions are not easily represented on a flat piece of
7'2
CHARTS AND GRAPHS
paper, you will find few charts which have at the same time
three axes.
Fig. 56. The Three Axes of Three-dimensional System of
Perpendicular Co-ordinates.
Students of trigonometry will recall another method of
measuring. They will say, standing at your point of origin
with the ‘‘:^-axis” rod in your hands, do not walk forward, but
turn slowly around. The points on the rod will then describe
a circular movement about you, and you can locate any point
Fig. 57. Polar Co-ordinates.
in the field by noting the distance on the rod and the angle
through which it has been turned. Indeed, you will find many
a chart which is built on this principle. Later on when we come
to some examples of it, we will have to explain its peculiar
qualities. For the present it is enough to say that the method
of “polar co-ordinates” can be used.
CO--ORDINATES
73
The great mass of chart work is built along plain co-
ordinates. The simplest form of all uses but one axis, and
consists only of a straight line; the commonest form uses two
axes, and consists of square or rectangular outlines. Occa-
sionally we have use for three axes, using three dimensions,
and we must not forget also, the circular two-dimensional
device with polar co-ordinates.
Ail this is so extremely simple that we hesitate to dwell so
long upon it. But unfamiliar names are great bug-bears; let
the idea be as simple as pie, technicians will come along and
give it a long high-sounding name, generally created on the
spur of the moment by themselves out of ancient Latin or
Greek dictionaries, but which even the old Romans and
Athenians themselves would have been unable to understand.
We must not let them fool us with these long empty names.
On the contrary, when we see ‘‘co-ordinates’^ we will know
that it means nothing more than criss-cross lines. We will
remember the criss-cross checker-board arrangement of roads
in American cities, and unless we hail from Boston, we will
think of “street” when we see the word “abscissa” and we will
think of “avenue” when we see the word “ordinate.” The
“;r-axis” is “Main Street” from which on both sides the other
streets f abscissae) are counted; the “y-axis” is “Main Avenue”
from which on both sides the avenues (ordinates) are counted.^
The point is that under no consideration will we let ourselves
get disturbed by the unfamiliar names, and in a little while
we shall be able to swing these words around with the best of
them.
2 “Main Avenue*' (the “y-axis’*j is crossed or cut by the streets (abscissae —
Latin for cut-ofFs} and “Main Street" (the “.v-axis") is crossed or cut by avenues
(ordinates — their Latin failed them here — ordinates means arranged in order). More
than that^ if the streets run east and west, and the avenues north and south, then
positive A is east along a street from Main Avenue and negative v is west; positive y
is north along an avenue from Main Street and negative y is south. Upper Central
Park West, for example, is negatives; from Fifth (Main) Avenue and positive y from
Battery Park.
Chapter VIII
DIMENSIONS AND VARIABLES
While it is deemed essential for artists to understand the
nature of crayons, brushes, and like materials for their
work, yet we often observe that they make a great to-do over
their study of human anatomy, landscape scenery, and the
subject-matter of their work in general. In the same way we
who would illustrate mathematical facts must, it is true, mind
our “p’s” and “q’s” — ^which is to say, our “abscissas and
ordinates,” but these are merely the tools with which we work,
and most of our attention should be directed to our subject-
matter — the data or statistics we wish to illustrate. The closer
is our analysis and understanding of the nature of this subject-
matfer, the clearer and more accurate will be our graph or
illustration of it.
Because this book is largely a manual on the craftsmanship
of charting, and will therefore be mostly devoted to the exam-
ination of the various types and kinds of charts with which
you can illustrate statistical facts, we here take one last occa-
sion, before the curtain rises on the charts themselves, to appeal
to the reader, on behalf of his own common-sense, and urge
him to review carefully his analysis of his statistics, before
laying ruler to chart.
We make this appeal at this time when the reader has
finished his statistical work and is addressing himself to its
charting. This is the time when, once more, he must stand off
and scrutinize the statistics which he has compiled in the face
of such great difficulties— the statistics which are as dear to
him as his own thought-children — and scrutinize them with
cold, calculating, dispassionate eyes. Do not make the mistake
of plunging into the charting-work direct from the statistical
work, but once more carefully weigh those statistics against
the original question, “What do I want to know?”
DIMENSIONS AND VARIABLES
75
The reason for this is two-fold. In the first place, this final
examination will give you valuable suggestions as to what part
of the statistics the chart should emphasize. It will enable
you to eliminate from your chart entirely what is not significant
to your inquiry, though the same may be necessarily retained
in the table for reference. It will enable 3?'ou to focus the chart
upon the essential information and present it to others in pre-
cisely the light or relation in which you wish it seen.i In short,
it will take the ^^straw'' out of your graph and leave only the
wheat.
In the second place, you need this last-minute scrutiny’' of
your figures, or some similar analysis when your results have
come clearly into view, to determine how you will chart the
statistics. Though it will not always decide the precise form
of chart for you, it will settle the fundamental principles of
that chart, and you can easily modify details later. The re-
mainder of this book will be devoted to the details of the
various charts, but here and now let us attack the fundamental
principles.
You have seen in the last chapter that measurement of
space is based upon linear distance in each of its dimensions.
A one-dimensional space would require but one axis or direc-
tion for measurements. A two-dimensional space would
require measurement in two axes or directions. A three-
dimensional space requires measurement in three axes or
directions. In short, there must be as many axes for measure-
ment as there are dimensions.
Now in precisely the same way, your statistical data can
be considered as having one, two, or more dimensions. If
there is but one dimension in your data, you should use but
one axis or direction on the chart-paper to picture it. If there
are two dimensions in your data you can use several one-
dimension charts or one two-dimension chart. If there are
three dimensions in your data, you can either present them
two at a time in several two-dimension charts, or try your
hand at one three-dimensional chart — a more complicated
form.
^ It is not intended here to suggest that it is possible or desirable to practice
d-ceprion, with any correctly-made chart, but only that the chart-maker can bring
out various asp.x'ts of the truth about his data with greater or less clearness, by his
choice of charting method.
76
CHARTS AND GRAPHS
How can you tell how main" dimensions there are in your
data? You can tell by the form which your data takes when
you have neatly tabulated it. You must allow one more
dimension to your chart than the figures to be plotted actually
require for correct tabulation. That’s the rule. For a figure
itself must be considered as having one dimension. (Imagine
each figure as reaching up perpendicularly off of the paper and
this will become clear to you.)
This rule can therefore be stated in another way. For
single figures use one dimension or axis on your chart. For a
series of figures, arranged either downward in a column or
across in a rowy use two dimensions. For a series of series of
figures, comprising a row or line of columns side by side, use
three dimensions. These last cases are not frequent, and are
limited to occasions where you could have presented the figures
in each row in a two-dimension chart, or the figures in each
column in a two-dimension chart, but wish instead to show the
figures by columns and rozes simultaneously, requiring, there-
fore, a three-dimension chart.
It is alwa3^s possible to present several charts in one. For
instance b^" using a number of one-dimension charts, 3"ou may
adhere to the one dimensional principle, but show several at
the same time. A series of horizontal bars, for instance, is an
illustration of this. (But when the series is arranged with care
as to the downward axis also, these man\^ one-dimension charts
can sometimes be made into one two-dimension chart.) Or
you may use a number of two-dimension charts, thereby
adhering to the two-dimensional principle, but showing sev-
eral at one time. A number of curves on the same chart-field
illustrate this. However, where the base-lines are diflferenti-
ated with care, these can often be made into one three-dimen-
sional chart. Charts into which several single charts have been
compressed are called multiple charts.
This brings us to an important exception or modification
of the simple rule for dimensions. For it is necessary that you
distinguish between variables and other functions. A table
which includes two or more functions which are not variables
IS really not a simple table but a multiple table, into wdiich
several simple tables have been combined. Thus, if the stubs
of your tabulation (that is, the items at the left of each line)
form a variable, they should be counted as requiring a dimen-
sion on the chart. But if they are not values of a mathe-
DIMENSIONS AND VARIABLES
77
matical variable, they do not add to the dimensions of the chart
but merely form a list of the number of charts which may or
may not be compressed into one multiple chart. The same
considerations hold true of the ‘^column-headings^^ or ‘^cap-
tions” of the tabulation. The stubs or headings constitute
variables, when their arrangement is fixed by their mathe-
matical sequence, but are not variables when they may be
freely shifted about.
The question of whether you are plotting individual figures
(no variable, one dimension), or series of figures (one variable,
two dimensions), or series of series of figures (two variables
and three dimensions), is therefore a matter of whether the
arrangement of the component parts is fixed or not. If your
individual figures (together with their stubs) can be shifted
freely up or down in their places, you are only plotting several
single figures, and need to use but one dimension. If however,
their arrangement is fixed and dependent on each other, you
are plotting one series of figures and need two dimensions.
Likewise if your columns or rows of figures can be freely shifted
about among each other, you are plotting several series and
need only two dimensions, but if their arrangement is fixed
and dependent upon each other, you are plotting a series of
series and need three dimensions.
All of this may seem very confusing just at present, but as
time goes by and you become more familiar with the nature of
your figures, you will begin to understand the interrelation of
these figures and then you will be able to use the rule just
considered to determine quite arbitrarily in advance the type
of chart you will need for illustrating them to the best advan-
tage. We will therefore table this matter for the present, with
the purpose of returning to it later on with a fuller under-
standing.
A corollary of this rule, however, you can keep and make
full use of from the outset. That is, never to use more dimen-
sions than are necessary for your chart.' Do not use tw'o
dimensions when one will serve your purpose, nor three when
two will do. If, through an excess of zeal, you violate this rule,
and use more materials than are necessary, you will merely
defeat your purpose and confuse the reader of your chart.
Let us take a simple example. Suppose we wish to illus-
trate with a chart the relative sizes of two cities, with popula-
tions 5,000 and 10,000 respectively. Obviously the second
78
CHARTS AND GRAPHS
town is twice as large as the first. There is but one variable
present (none in the data-stubs or column-headings), the sizes
of the cities. According to our rule we will use one-dimension
charts for these figures, but as there are two of them, w'e will
have to use two charts.
Because the charts are single-dimension or single axis ones,
it is obvious that they will be straight-line ones. Two lines or
bars, the one of which is twice as long as the other, will illus-
Fig. 58.
trate the two populations, so that the reader cannot help
arriving at the conclusion that one city is twice as large as the
other, merely from a glance at the chart. This chart will be
both clear and accurate. It is therefore correct.
But suppose we had in this case used two-dimension charts.
Suppose for example that we had used square areas instead of
lines. And because the second city is twice as large as the
first, suppose we therefore made each dimension in the second
□ I I
Fig. SB*
chart twice as great as the corresponding dimension in the
first chart. You will already be objecting that the area of the
second chart will not be twice as great as the area of the first,
but four times as great. Look at the chart. Notice how sur-
prising is the difference between the two cities as here repre-
sented.
And notice also that it is very hard, from the chart alone,
to decide exactly what the ratio between the two areas is. The
eye does not readily compare areas with precision. At first
glance one might be inclined to say that the second area is
five times as great as the first; though we who made it know
that it is only four times as great. This illustrates a very
important rule — that it is difficult to compare areas. That
rule follows from the original principle, not to use more dimen-
sions than necessary, but it is worth keeping in mind as a
particularly important phase of the principle.
DIMEiVSJONS AND FARIJBLES
19
But you will say, “Cannot we show the areas in true pro-
portion?” We can, if we will but take the trouble. A little
figuring will show you that if the areas are to be in the ratio
of 5,000:10,000 or 1:2, the sides of the squares must be in the
ratio of one to the square root of two. Get out your pencil
and paper — or your slide rule — and figure out the square root.
It happens to be 1.414. Consequently, if the square areas are
to be in the proportion of one to two, the sides of the squares
□ u
Fig. 60 .
must be in the proportion of 1 to about 1.4. Now draw the
two square charts in such a way that the sides of the second
are 1.4 times as long as the sides of the first.
A glance at the resulting chart will disappoint you. Instead
of making the difference in size clearer, you have apparently
minimized it. The two areas no longer seem so very different
in size. It will take a clever reader indeed to realize from a
study of the two areas that the second city is fully twice as
great as the first city. And the worst of it is that many
readers will, through ignorance, fall back on the length of
the sides as a basis of judgment, and decide that one city is
only half as large again as the other. In short whether you use
the sides or the areas as your own basis, there is a good chance
that your reader will happen to use the other basis and so be
entirely misled by your chart.
Obviously this is no less true when we use circles or other
regular areas in the place of squares. A circular, like a square.
area varies with the square of its linear measurements. If
you make the radius of one circle twice as great as the radius
of the other, the first area will be four times as great as the
first. If you make the areas proportionate, the radii must be
in the relation of 1 to the square root of 2. Both circle and
square require the more or less tedious computation of square
8o
CHARTS AND GRAPHS
roots and repay this labor with inaccurate and ambiguous
results.
o O
Fig. 62.
We can carry this example further, into three-dimensions.
That is to say, suppose we attempt to show a one-dimension
fact by a three-dimension chart. Suppose some bright young
illustrator suggests that, since it is population we are showing,
we use the picture of a human being for a chart. Now let us
make the height of one person just twice the height of the
other. What has happened to the volume or W'eight of the
two persons — assuming that they are similarly shaped :
Clearly one is eight times as large or as heavy as the other, for
the volume of cubes or solid bodies varies with the cube of linear
dimensions. Our pictures give a grossly exaggerated impres-
sion of the comparison of the two cities.
To counteract this, we can make the volume or weights of
the two bodies represented in the proportion of one to two.
Fig. 64.
DIMENSIONS AND VARIABLES
8i
But in this case their heights and other linear dimensions are
going to be in the ratio of one to the cube root of two. You
will find if you look it up, that the cube root of two is 1.26.
And you will be even more disappointed with your resulting
volumes when you have drawn their heights to scale, than
you were with the resulting surface areas. Meanwhile your
poor reader will be trying to choose between three ways of
judging from your picture — height, surface area, and cubic
volume — ^with a two to one chance of making the wrong choice.
The illustration of population by areas or solids is a fre-
quent type of faulty chart. We have all seen pictures pur-
porting to show the sizes of various armies, for instance, by
pictures of soldiers drawn to different sizes. Even the United
States Census,^ has turned out an entire volume of several
hundred pages, containing many circles illustrating relative
sizes. It is perhaps the commonest form of deceptive or
ambiguous chart used — most frequently occilrring when the
author has tried to combine a picture of his items with a chart
of their mathematical ratios.
When you see such a chart in the future, learn to ask your-
self what part of this chart shows the true ratio, height, area,
or indicated volume. In nine out of ten cases you will find,
by looking at the data, that the height or linear dimensions
are in true proportion. You will then realize that the area or
indicated volume is grossly deceptive and you will not be
Fig. 65. The Effects of Comparison by Linear, Square,
or Cubic Measures.
2 One of the greatest official accumulations of charts is the Statistical Atlas of the
Twelfth Census of the United States, in which circular areas were used very exten-
sively with correct interpretation of values by areas.
82
CIURTS AND GRAPHS
misled by it. In any case you will quickly see that the man
who made the chart either did not know his business or if he
did, was guilty of shameless intent to deceive.
In short, the rule that no more dimensions or axes should
be used in the chart than the data calls for, is fundamental.
Violate this rule and 3’ou bring down upon 3’’our head a host of
penalties. In the first place, you complicate your computing
processes, or else achieve a grossly deceptive chart. If your
chart becomes deceptive, it has defeated its purpose, which
was to represent accurateh". Unless, of course, you intended
to deceive, in which case we are through with you and leave
you to Mark Twain’s mercies. If you make your chart accu-
rate, at the cost of considerable square or cube root calculating,
you still have no hope, for the chart is not clear; your reader
is more than likely to misunderstand it. Confusion, inaccuracy
and deception always He in wait for you down the path depart-
ing from the principle we have discussed — and one of them is
sure to catch you.
Chapter IX
HUNDRED-PER-CENT BARS
The division of a “whole” into its “parts” is logically one
of the first steps -in any analysis. Usually the graph illus-
trating this division belongs at the beginning of a statistical
report. Thus, if your report covers the sales of the company,
your first chart would break up total sales into the individual
sales for each line or for each district. The remainder of the
report, treating of details of the various “parts” (e.g., lines or
districts) will then follow a summary chart which has estab-
lished their relative importance.
A quantity can always be illustrated by a straight line, or,
as it is commonly called, a “bar.” Bars are the simplest and
often the best form of graphs. The total length of the line
then represents the total value of the quantity. When we speak
of a line in charting, we do not mean an imaginary straight line
having neither width nor depth, for that would be invisible
and could not, of course, be actually used in illustrations. In
its place we use the bar, with a visible width (and the actual
depth or thickness of a layer of ink). But it is still proper to
speak of this bar as being a line or one-dimension chart, for its
width and thickness are constants, necessary to give visibility
to the line, and its length alone is significant.
Now a single bar, illustrating a single figure, will have no
particular meaning for the I'eader, because he has nothing to
IBZQ
Fig* 66, A Simple 100^ I Bar,
84
CHARTS AND GRAPHS
compare it with. There is therefore but one case in which we
have any use for the single bar, shown by itself. This is the
case in which we wish to show the parts into which a total
may be broken up, or of vrhich it may be composed. And
whatever the quantity or total be which is thus composed of
two or more parts, we may call it 100 per cent of itself and the
parts will then be certain percentages thereof. Hence, this
single bar shown by itself, has come to be called the ^T00%
bar,’’ regardless of whether the total and part quantities be
quoted in the data and chart in actual values or in relative
percentages. Irrespective of scale or calibrations, the bar rep-
resents a whole or 100 per cent, and its divisions or parts rep-
resent parts or percentages thereof-
I?*
A*
o.4;ub ifi.s *»:
PERIODIJAW IH the ITiItiD STAI»
IS.’O
(Koure* - H. Tl. Ay dr k B«>)
Fig. 67. Many Segments, No Shading.
In the 100% bar we have the mathematical equivalent of
the classification chart. Turn back to Chapter III if you
have forgotten what a classification chart is — or better still,
don’t turn back, but stop and try to remember it. The classi-
fication chart sets forth the ideological or schematic relation
of things. Commonly it displays the parts of which a whole
is composed. When it does this, it is very similar to the 100%
bar. The difference between the two lies in the fact that the
classification chart shows what the parts are but does not tell
their relative importance or size, while the 100% bar shows
their relative importance or size, upon the basis of certain
numerical data.
Thus the best labelling (for labelling is half the work of
chart-making) for the 100% bar would seem to be a classifica-
tion chart placed, obviously, above the bar. As a matter of
fact, wherever it is typographically possible, this is correct.
Use a classification chart to label a 100% bar, or a 100% bar
to illustrate a classification chart, and you have the best pos-
sible combination. Of course where some parts are relatively
very small, typographical difficulties arise. If you cannot over-
come these by setting certain words on edge, you will be
HUNDREB-PER-CENT BARS
85
n
“L
Battle
Other
47,918
29,203
= 1 :
~1
Killed
Died
—
Died
in
of
of
Action
Wound B
lisease
34,218
13,700
23,430
Total Casualties
302,612
V<ounded
Severely
83,390
Wounded
221,059
Diaunded
Slightly
91,189
Degree
Unlmown
46,480
jOiERICjyS CJiSDALTIES IH THE TORLD KAB
1917-1918
Fig. 68. Classification-chart and 100% Bar*
obliged to content yourself with a general grouping of minor
parts into a single part labelled “Other,” “Minor,” “Miscella-
neous” or some such rag-bag title for stray odds and ends.
A further detail of the 100% bar and its labelling, is the
scale. This should generally be in hundredths or percents.
The data may be entirely in absolute quantities, but neverthe-
less the scale should show percentages. To prevent the con-
fusion of scale and divisions of the bar, the scale should be
outside the bar, and the best practice seems to be to indicate
the scale by little notches or short perpendicular lines dropped
below the bar, from its lower edge. The scale should have
ten, twenty, or a hundred of these little lines, each indicating
a division of ten, five, or one per cent. The purpose is to enable
the reader, by counting notches on the scale, to compare parts
of the bar with greater accuracy. For the same reason, the
actual percentage to which each part is equivalent should be
written or printed below the bar under the center of each part.
The bar itself, as has been said, should be of appreciable
thickness. Too light or narrow a bar, such as a thin line, has
no emphasis or force. But too wide or heavy a bar introduces
two-dimensional rather than linear conceptions in the mind
of the reader and sometimes produces undesirable optical illu-
sions. The width of a bar should be such as to make it clearly
visible at the distance from which it is to be viewed. The best
86
CHARTS AND GRAPHS
form of bar is generally between a tenth and a twentieth as
thick as it is long.
The bar should be hollow, that is, outlined. The segments
or parts of the bar may also be hollow, but it is better to shade
them with distinct colors or shadings. Small dots, various
hatchings (cross-lines), and double-hatchings (criss-cross lines),
can be used to distinguish the various parts without using
colors. But where colors can be used, they are sometimes de-
THS FAHltY BCCOST
«• to Clas»«s of Comoditl**
Uni tod StAtfls
1913
(Soure* - Monthly Lohor
(Hot#; ViL, Pu#i nfii LiRhUnj, ?4F. Furr.itur# *aci FurniihlniEi J
Fig. 6S. Distinct Shading.
sirable, for the reason that solid shades are more forceful than
black and white shadings. Care must be taken in either case,
however, to see that the various parts are similarly emphasized
by the color or shade, for otherwise one part will appear more
important or larger than it really is. A solid black area will
appear to be larger than a solid white one outlined in black,
though really of exactly the same size, for the black is in itself
so much more powerful than the white, and has further gained
by absorbing its black outlines. Experiment will soon show
rmiOK tbauc catpiays
U 20
Fig. 70. Two Bars are Easily Compared.
HUNDRED-PER-CENT BARS
87
whether an optical illusion^ is being introduced by the shadings,
and those combinations which will bring out the various parts
equally.
The data for the 100% bar need be no more than a list of
the parts of which the whole is composed, with their respective
percentages, and either with or without their respective abso-
lute quantities, according to your wish. If the quantities are
important, or you think that someone is likely to call for them,
add them and forestall criticisms; if they are unimportant,
they can be safely omitted. While the percentages are almost
always desirable and are best placed below the bar, as part cf
the scale, the absolute figures or data, if inserted, are best
placed immediately over the bar, as part of the classification
chart which is used for the labelling.
That data of this character calls only for a single dimension
Year 1913
ThE FAMILY BUDOET
OiTidad as to Claaaas of Conaodltiot
United State*
1913, Dec 1920> Deo 1921
(Source, - Monthly Labor HevidJf)
(Note:- tkl, fuel and Lighting; F4F, Furniture and rurnl»h£nj;l)
Fig. 71. Comparison of Three Different Years.
chart, according to the rule of chart dimensions is obvious.
For the individual figures, with their stubs, in the table of
data, can be shifted freely up or down the column or across in
the row in which they are tabulated and the stubs therefore
cannot be said to form a variable. In another chapter we shall
consider another way of presenting the same data, in which
each figure forms a separate bar, and the series of figures in
^ For a discussion of the various optical illusions to be avoided, see Willard
C. BxmtonyGraphic Methods for Engineer ing Magazine Co., pp. 358,
359. See also Appendix I).
88
CHARTS AND GRAPHS
the data are shown by a series of separate bars. In the 100%
bar you may, if you wish, think of these same bars as being
again showm separately, only placed end to end instead of
one above the other. And data of the 100% type, that is, in
which the figures can be added together to form a coherent
and significant total, is the only case of data which can be
shown on the 100% bar. For all other series of figures, the bar
charts discussed in a later chapter must be used.
Chapter X
PIE-CHARTS
Throughout your study of charts you will find some which
are more useful for popular consumption than others, but you
will not find many which are more purely popular in appeal
than the 100% circle or pie diagram. For analytical purposes
it has nothing to recommend it, but for sensational values it
is in general without an equal. If you are research-bent, you
may safely pass by this chapter on popular exegesis, but if
your object is advertising, you will seize it to your heart.
5 We have just seen how a single bar can be taken to repre-
sent 100% and can be cut up into segments or parts the lengths
of which correspond to the relative sizes or percentages of the
various parts of the 100%. The fact that the line is a unit,
and so long as it remains the whole, can never be more than a
unit or 100%, should suggest something. It should suggest
that the total length of the whole line is relatively unimportant.
It is unimportant because the reader is not asked to compare
the total length of the line with the total length of any other
line. There is no other line to compare it with, unless a second
100% bar is lying around handy, in which case the second
would presumably have the same length, because it too repre-
sents 100%. Hence, you will say, why have any total length
at all.^
Centuries ago it was a moot question among philosophers,
whether the Lord could make a yardstick which was endless.
Then someone suggested that the yardstick be bent into cir-
cular form and the question was dropped. Let us perform the
same operation on a 100% bar. Imagine, if you wish, that
the bar is so very thick for its length that while one edge be-
comes the circumference of the circle, the other shortens down
to and becomes the center of the circle. Division lines between
the component parts of the bar become rays or radii of the
circle and serve to mark off the corresponding component seg-
8p
9°
CHARTS AND GRAPHS
ments of the area of the circle. Here you have in a nut-shell
the pedigree of the pie-chart.
IMPORTS INTO RUSSIA
1921
(Source:- Russian Information and Eeriew, London)
(Grand Total, ♦124,281,000)
Geraiany
Fig. 72. A Simple 100% Circle*
It is now time to let you into the secret that the rule of
dimensions of charts, which you doubtless memorized in a
previous chapter, has apparent exceptions. The pie-chart is
one of them. For few readers will judge quantities by either
the arc at the perimeter of the circle or the subtended angles
at its center — on the contrary most of them will judge entirely
by the areas of the segments. In short, the pie-chart appears
to be a two-dimension (area) chart used for one-dimension
data. The fact is, however, that, as in the case of the 100%
bar, the area of the chart varies directly with one dimension,
the other dimension being constant. In the 100% bar the
width of the bar was constant in the 100% circle the radius
must be constant for all circles compared. Then the area*‘of
the segments varies directly with their arcs or angles and the
chart has but one significant dimension. It is only an ap-
parent exception to the rule.
PIE-CHARTS
9T
The disadvantages of the pie-chart are many. It is worth-
less for study and research purposes. In the first place, the
human eye cannot easily compare as to length the various arcs
about the circle, lying as they do in different directions. In
the second place, the human eye is not naturally skilled at
comparing angles — those angles at the center of the circle,
formed by the various rays or radii and subtending the various
arcs. In the third place, the human eye is not an expert judge
of comparative sizes of areas, especially those as irregular as
the segments of parts of the circle. There is no way by which
the parts of this round unit can be compared so accurately and
quickly as the parts of a straight line or bar. Moreover, when,
as frequently happens, several pie-charts are shown together,
the various slices in one chart cannot be so easily compared
Fig. 73- Accurate Comparisons Cannot be Made.
92
CHARTS AND GRAPHS
with the corresponding slices in the next, as can the various
parts of one 100% bar with corresponding parts of another
bar. The two bars can be placed one above the other, so that
comparison from one to the other can be made at once, but
no arrangement of the two circles will make comparison so
simple.
In the labelling of the pie-chart, you will furthermore en-
counter typographical difficulties. It is not ordinarily a good
thing to make a reader crane his neck at various angles to read
writing along every point of the compass, so you should not,
as so many do, write on radii from the center of the circle.
On the other hand, unless the chart and its segments are very
large as compared with the size of the printing, you will intro-
duce tricky optical illusions if you write all labels in the same
directions inside the segments.
mCHASIBO POWK Of THf OOiUE
Of 1913
lAmn tt<*d for food rotail
0. S*
Fig. 74. The Less Detail, the Better.
PIE-CHARTS
93
Sometimes, the best rule Is to put the labelling away In a
key or explanatory list of the shades or colors of the various
segments. Only if the labels are very brief, and your segments
are all large, can you stow the labels into the segments without
greatly altering their apparent sizes. When neither plan is
feasible, and you feel that you must have each segment, how-
ever small, immediately labelled, place the labels outside the
circle, adjacent to the proper segments, with the printing in as
nearly the same general direction for all as you can arrange,
so that the apparent sizes of the segments are not confused
by printing and the reader need not climb around the edge of
your chart to decipher it.
As a general thing, however, there is one part of the label-
ling which can always be attached to the chart, namely the
figures of the percentages for each segment, which in the bar-
Retail Food Establishments
New York City
1921
(Total Number, 75 , 412 )
9 ^
chart were placed immediately below the bar. These figures
should always be placed close to the segments, but usually out-
side the circle, so that the reader who wishes to have the precise
percentage figure represented by a segment, can always do so.
The scale (without scale-figures) may be placed inside the
circle and unlike the 100% bar may or may not show in the
finished chart. Special paper is marketed for these charts with
94
CHJRTS AND GRAPHS
PIE-CHARTS
95
the circle printed in and already divided into a hundred parts
by small notches within the circumference. The use of this
paper will save you much time if you wish to make the seg-
ments accurate in size. It is tiresome to use a protractor
marked off into 360 degrees, and to calculate the decimal
equivalent or percentage in degrees. Unfortunately, the metric
system has not been applied to circular measurement so as to
give any circular or angular measurement which employs a
decimal system.
96
CHARTS AND GRAPHS
The advantage of the pie-chart is psychological. It in-
stantly commands the reader’s attention. A circle is, of all
geometrical patterns, the easiest resting spot for the eye. The
fact is well known to advertisers, who frequently use circles
and circular outlines to draw attention to their advertisements.
Hence if your chart is designed for publication, or for presenta-
Bisect 01) at E and project EF equal to EA. Project the chord AG equal to
AF. With dividers set for this distance, lay off from A the points, G, H, K, and
J (as in upper right-hand diagram). Similarly lay off four points from B, C, and
D (as in lower left-hand diagram). Erase all other marks and calibrate the
twenty points so found, to form the finished circle (as in lower right-hand diagram).
tion to readers whose attention may be easily diverted, you
will find the pie-chart a powerful means for presenting your
facts. Attention will be focused upon it at once, and it is as
PIE-CHARTS
97
simple to understand as its name — far too simple for anyone to
misunderstand. Because it is circular, there is no question
but that it represents a whole and the various slices of the pie
belong to their respective items.
Cost of the World War to the United St.ates
Estimate on July, 1521
Grand Total Cost — 350,168,625,707.16
(Source; World Almanac)
A very sound use of the pie-chart occurs in the case of
financial data. Here the whole circle or pie can be spoken of
as a ^^dollar/’ and the various segments, the parts of the dollar,
so many cents (or percentages) each. Charts of this type have
been used to show the distribution of costs in a plant, or the
parts of a financial budget, or the shrinkage of the dollar in
high-cost-of-living studies. Through the fortunate coincidence
that our metal currency is round and our dollar divided into a
hundred cents, the shape and the labelling of the chart both
find immediate understanding in the mind of the reader.
Fig. 80 . A Pie-chart in Metal.
The ‘‘Swift Dollar,” as it was called, on one side of which a chart shows the
division of income from sales.
The pie-chart must therefore be accepted as an advertising
medium of value. It has strong popular elements. But it has
98
CIIJRTS /JND GRAPHS
no place in the statistical workshop, or research laboratory.
Before using it in the place of the simpler and sounder 100%
bar, you should carefully gauge your audience or readers, and
only if you believe that you have begun to strain their interest
should you judiciously insert the pie-chart. In a sense, it
might be construed as an insult to a man’s intelligence to show
him a pie-chart, but the insult is not often resented. For if
your main object be to get a story across, you are justified in
taking that means which will encounter least resistance, and
in making your story as simple as possible. For publicity
purposes, the pie-chart is therefore almost invariably better
than the bar.
Chapter XI
BAR-CHARTS
A senes or quantities or values can be most simply and
often best shown by a series of corresponding lines or bars.
All bars being drawn against one and the same scale, their
lengths vary with the amounts which they represent. In a
previous chapter, the 100% bar was described, in which a
single bar, whose total length had no significance, was divided
into parts in order that the relative size of the parts nfiight be
seen. In this chapter we propose to use several bars, which
are not divided up into parts, but which can be compared as to
their total lengths, in order that their relative sizes may be
seen. While this new method could be employed with the
same data, it is generally more useful for data in which the
various items are not being shown as parts of a total, but as
individual and co-equal totals in themselves,
BUSINESS FAILURES
Amount of Liabilities
United States
1920
(Sources- United 'States Census)
(Millions of Dollars)
Bar-charts are most flexible and can be varied to suit the
individual whims of the maker. ■ In general, however, there is
one style or form which will be found most satisfactory. It
consists of a horizontal grouping of bars alongside of the data.
The chart is arranged in tabular form, with items or stubs in
99
lOO
CHARTS AND GRAPHS
a column to the left, with figures in a column beside the stubs
and with bars in a column beside the figures. Several columns
of figures are sometimes desirable, just as in the table of data,
to show sources or original figures from which the charted
Population
Arsa
square miles
Pop,
per
sq.mi <
TOTAL
1,702,520,000
57,255,000
20,6
Buropt
464,661,000
3,873,000
120,0
Asia
872,622,000
17,206,000
50,7
North America
160,000,000
8,689,000
16.3
Africa
142,750,000
11,623,000
12,3
South Amerloa
56,340,000
7,570,000
7.4
Australasia
16,230,000
3,315,000
4.9
Polar Regions
6,082,000
(Seal* of Population por S.o, Milo)
DENSITY OP POPULATION OP THE BARTH
by continonto
Fig. 82 . Detailed Data may be Included.
figures are obtained. In any case, the bars should represent
the most important set or column of figures, and there should
be normally but one column of bars.
There should be but one column, of bars because the bars
can be advantageously compared only when they are side by
side, one below the other. Thej" should all begin at a uniform
point or distance from the figures, so that their lengths can be
compared out at the far ends. They should be the last column
on the page, because their uneven lengths make further col-
umns wasteful of space, and the addition of further columns
introduces optical illusions which should be avoided. If a
column of bars were to be followed by a column of figures,
the reader would be apt not to compare the lengths of the
bars, but their shortage from the last column.
It is a very common error to place the data inside the bars,
for by so doing the reader is led to compare those parrs of the
bars which are clear of figures, rather than the entire lengths
of the bars. This optical illusion exaggerates the difference
in lengths of bars. Another mistake which is often made is to
place the data out at the varying ends of the bars, for here the
reader is led to compare the lengths of bars plus data, rather
than of bars alone. Here the optical illusion minimizes the
BAR-CHARTS
lOI
difference between bars. The proper place for data is in a
column at the left of and immediately before the bars them-
selves, with no more reading matter to the right of the bars.
The scale for a bar-chart should be placed at the top of
the chart, immediately above the uppermost bar. A field in
fainter color (green is most useful for chart-fields) or thin lines
should be drawn into the chart by extending down from the
scale a few lines which mark off convenient distances on the
scale. Thus the reader is enabled to compare lengths of bars
far distant, by noting their relative positions against the field
or background. Care should be taken that the lines of the field
do not cross the bars, else the field will cease to be a back-
ground and will become a screen in the foreground.
(Source:* Bepori of Seiuttoriel Coffinittee)
Figr. 83. A Long Bar Broken to Save Space.
The bars should be of uniform thickness or width, as it is
the variation in their length which is significant and variations
in width would produce area-illusions. This rule is obvious —
so obvious that where a large number of bars are shown and
the reader is already thinking in terms only of the lengths of
102
CHARTS AND GRAPHS
the bars, it is permissible to violate the rule to emphasize im-
portant or group-total bars. The simplest example of the
extra-wide bar is an average or total bar for the entire series.
Fig, 84, National Distribution by States and State-groups.
BAR^CHARTS
103
The extra width given this bar need be only slight to make it
stand out from the rest as a sort of type or normal against
which the individual bars can be measured. And this should
be done only when there is no danger of the reader's attaching
importance to areas.
It sometimes happens that you desire to show two tables
on the same subject, one giving a large number of individual
items and the other a few sub-totals. It is a good plan in
such cases to place the two on separate sheets that face each
other, so that they both show at the same time, the one giving
summaries of the other in the form of sub-totals. Where the
summaries are not averages, but are true summations or
TRADE DKICaiS OP THE WORLD
Xitijnated M«mb«rohip of trade tmione in twenty oMef oountriea
1919
(Source:* International Later Office)
TOTAL *32,680,000
Australia
628,000
■*1
Austria
772,000
—
Eelgiuai
760,000
Canada
378,000
□
CteehosXoTsikla
667,000
....
DeiBBark
360,000
□
Finland
41,000
1
France
2,600,000
□
Oermany
2,000,000
1
Great Britain
8,024,000
Hungary
600,000
-
Italy
X, 800,000
Wetbsrlands
626,000
P
New Zealand
100,000
5
Norway
144,000
D
Rounianiu
tmknown
Serbia (Jugoalavla)
20,000
1
Spain
211,000
□
Sweden
330,000
□
Switzerland
224,000
□
On! ted States
6,607,000
Cl
4 =-
lj
Ti
Fig. 85. An Alphabetic Arrangement#
104
CHARTS AND GRAPHS
totals of the items they include, you will find it well-nigh im-
possible to draw both charts to the same scale, without having
one or the other so large or so small as to be useless. You will
therefore be obliged to shorten your scale, that is, use a smaller
scale for the summary-bars than for the individual bars.
In such cases it is not a bad plan to make the sub-total
bars wider. In exactly the same proportions as you condense
the scale, you should thicken the bars plotted thereon, so that
the areas of the summary-bars will be equal to the combined
areas of the individual bars of which it is the total. This is
one case in which a slight use is made of the conception of
area, or two dimensions, but it is negligible, for the reader is
not called upon to compare areas — only lengths — in all cases,
and the thicker bars remind him that the scale has been short-
ened on the sub-total chart and prevent his making confusing
comparisons between the two charts. In short, when the group
item in a series is an average of individual items in the series,
it can be shown on the same scale, but where it is a total or sum
of individual items, it can not be shown on the same scale
without making the individual items small, but can be shown
on a separate chart in reduced scale and with correspondingly
increased thickness.
1800
1810
1820
1850
1840
1860
1860
1870
1880
1890
1900
1910
1920
Fig. 86.
(Scale of Dollars)
PJKCAPITA PUBLIC DEBT
iesa Cash in Treasury, July Ist
United States
1800-1920
Historical Data Must be in Order.
BAR-CHARTS
105
The arrangements of items in a bar-chart should be a
matter for careful study, and no arrangement should be chosen
which is not the best adapted for the special purposes of the
chart. It is impossible to lay down absolute rules, for each
case varies, and the author of the chart must rely on his own
judgment rather than on rules of thumb. But in general the
principles explained in a previous chapter on work sheets
apply here, as the bar-chart is closely akin to the mere statis-
tical table. When the items are historical, the earliest dates
should be at the top and the last at the bottom of the chart,
IHE OADSES Of HSES
United States
1915-1919
(Source:* Kat‘l Board of Fire Understitora, N» Y.)
(Value)
C
TOTAL 1,135,000,000
Eleotrloity 84,100,000
Matches and smoking 73,500,000
Uefeotlve chimneys 66,700,000
Stores, hollers & pipes 66,100,000
Spontaneous ocmibuetlon 49,700,000
Ughtning 39,600,000
Sparks frcaa machinery 31,900,000
Si%rks ott roofs 89,300,000
Fetroletm h products 86,900,000
Sparks froa oomhusiion 86,100,000
Inoendlarlaa 21,600,000
Open lights 14,000,000
Ashes, coals A open fires 11,810,000
Oas, natural le artlfio* 10,200,000
Jbatplosions 10,160,000
Hot grease, oil, tar,eto 4,490,000
Kubhlsh and litter 3,610,000
Steam k hot -water pipes 1,860,000
Fireworks, fireoraoker# 1,600,000
Fig. 87 . Placing the Most Important First.
the order being strictly chronological. When the items are
geographical, as for example a list of the States in the United
30ALE OP MILLIONS OF DOLLARS
10 20 30 40 60 60 70 80 90
CHARTS AND GRAPHS
io6
(ScaX« af fitLllions «f AfiniWn^
0 12 3 4 5 6
Cotholle
1T,S49,824
r, *, (whtto)
6.328,476
r^ptict (whiu)
4,389. T69
Baptist A M. 1. (colored)
4,191,267
Luthoran
2,451,997
Prtabytorlan
1,603,033
Ditolploi of Christ
1,193,428
Proto stont Episcopal
1,066,828
Congrogatlenal
808,122
Itormont
494,388
StLIQlOUS DSNOiailATIONS OF UNITED STATES
Mrabervhip
1919
Sourea:* "Year Book of Churches*
Fig« 8S* The Arrangement in Order of Size is Popular.
States, a geographical arrangement (either in the order given
in the Census volumes or in some specially designed order) is
. of Central Compef/fm fiald _ __ _ _©I7
[
STATE
details OF IDLENESS DUE TO
MONTH
USED OK , SECQNllMALF YCAH
10 20 50 40 SO eO 70*80 'bo
UCK OF
WORK.
(cars)
LACK OF
HELP
LACK OF
AND POOR
MATERIAL
REPAIRS
POOR
lAHNINS
REMARKS
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Fig. 88. The Cantt Idleness Chart.
Showing the failure^ to produce to capacity, together with data analysing the
failure. This chart is one of a series of such charts on the coal mining industry,
appearing m The Dialr-Refroduced from ''The Life and Work of Henry X.
Gantt^' American Society of Mechanical Engineers, paper of Mr. Polahov,
BAR^CHARTS
107
preferable. A mere alphabetical arrangement of the States
would have little to recommend it, as the reader of the chart
must be presumed to have sufficient intelligence not to require
a dictionary of the country. The popular arrangement is in
the order of magnitude of the data presented, so that the
longest bars are at the top, but unless the reader’s sole purpose
is to glimpse the names of the leading States, this arrange-
ment is useless, as it lacks comparability with other charts.
In general the arrangement of the items should be such as to
afford the greatest aid to such analysis and comparative study
as the chart may be subjected to.
Fig. 90^* Classification-chart and Bar-chart «
BARNHARTS 109
serted under the heading of the data which is plotted to show
to which figures the bars belong. The scale of the bar-chart,
above described, forms the heading for the bars themselves.
When a series of charts are shown together in the same report,
the charts of absolute figures can be displayed in typing and
bars of one color, such as black, and the charts of relative per-
centages, per capitas, or averages can be distinguished by
typing and bars of another color, such as red. In general a
judicious use of colors will assist the column-headings and
titles of the charts in differentiating charts in large sets.
The technique of bar-charts is so simple and they are so
very effective, that they should be used freely in printed text-
matter. No drawing or plates are needed. Printers have
'"rules’^ as they call them, which can be used to make solid
bars, and these rules can easily be set up together with the
type. The scale and field can be omitted and the bars alone
will effectively tell the story of the main figures in the table.
The combined table and chart can be used in printed text just
as well as the table alone.
Fisf. 93. Bars as Part of the Office Record.
no
CHARTS AND GRAPHS
8«n
Ihilted lingdon
1.3d
Mgim
1.02
frustia
2.25
Auftrift
1.16
FATilLXTY BATS IK COAL HIKIMO
fatal Aeoidante par 1000 Worlcaaa
Spaoifiad Cotlntrlaa, 1919
(Souroa: Buraau of Labor Statlatlca)
Fig. 94. Typewritten Bars for Typed MS.
When manuscript is typewritten, the bars can, if desired,
be typed in, using such letters as “x” or better still, “x,” “o”
and “m” printed one over the other. This obviates the need of
drawing, and illustrates excellently any tables which you are
obliged to insert in the text. It has the further advantage of re-
producing on carbon^ hektograph, or mimeograph copies
(though the latter can be secured also in drawings by the use
of a special stylus).
^Not only can various combinations of colors in typewriter ribbons, be obtained,
but also various colors in carbon paper, which can be successively inserted in the
typewriting machine to reproduce the desired effect. Black and red are, however,
usually sufficient,
Chapter XII
COMPOSITE BAR-CHARTS
What man has done once, he can do again, and since we
have put several single bars together to make a bar-chart, we
can put several bar-charts together to make a compound or
multiple bar-chart. The single bar represented a single figure,
the simple bar-chart a series or column of figures, and the
compound or multiple bar-chart will illustrate a series of series
(or columns) of figures. For the sake of convenience, we can
divide these last into two classes, one of which we may call
the compound and the other the multiple bar-chart. Nomen-
clature is of little importance but a precise use of names will
help to distinguish two radically different forms.
Where each bar in a bar-chart is divided into parts, as was
the single 100% bar, the name compound bar-chart is sug-
gested. In such cases each bar is really a 1(X)% bar by itself,
BUSINESS FAILURES
Aaount of Liabilltio*
United States
1916-1920
(Source*- United State* Cenaue)
Hanu-
fac-
1916
1917 80
1918 73
1919 S2
IS20 128
Tra- Bank- Grand CUiHion* of Dollar*)
but its length may be no longer constant and uniform, and
may be made'to vary in the fashion of the simple bar-chart.
Its scale and labelling follow the same form. Data again
should be at the left of the bars. The scale should be above
the bars with a field projecting it downward behind them.
HI
xia
K
m
M
a
O
«Q
O
o
C4
o
M
«
»-4
A
CHARTS AND GRAPHS
Fig. 96. The Chart Does Not Suffer from Detailed Statistics Attached.
COMPOSITE BAR--CHARTS 113
The bars should be placed beside and on the same line with the
data which they illustrate. As to the arrangement of the
columns of data, it is best to have the column of totals, which
the entire bar represents, at the beginning or end, and then
beside it, in the order in which the parts will appear in the
bars, the columns of various parts. Shading of the parts will
follow the rules given in the chapter on 100% bars.
We will find two distinct types of the compound bar-chart,
each belonging to a certain type of data. If the data exr
presses the values in absolute quantities, the totals for the
various items or stubs (lines) will not necessarily be equal — in
fact will very rarely be equal. So that the chart of this series
will have bars of unequal total lengths. (Needless to say, the
widths of bars will always be constant and uniform, save for
the special exceptions noted in a later chapter.^) Such a chart
has its scale and field measured off in units of absolute or
COAL
Total all
kinds
OIUND tOTAL
MO
7,460,506
trnit«d Stat«t
S, 538,506
Cajutda
1,561,000
Chlaa
1,097,000
0»naBXQr
467,000
Vrsat Britain
209,000
Sibaria
192,000
Australia
185,000
India
87,000
Bussla (in Buropa)
66,000
thtion of S. Afrioa
62,000
Austria
59,000
Coloaibia
30,000
Iaido*ChlBa
22,000
rrutoa
19,000
Balgim
12,000
Spain
10,000
Spitxbarsoa
9,000
iVapan
9,000
Solla^
‘i’ooo
Oibor Countrias
24,000
Fig. 97. Very Small Segments May Be Shown.
^ Cf, Chapters L. and LL
CHARTS AND GRAPHS
114
actual quantities, such as dollars, pounds, tons, or whatever
the unit quantity be in which the data is expressed. It is an
“absolute” chart, or chart of absolute values.
The other type of data is derived from the last, but is
often more significant. In it the quantities have been turned
into percentages, in each case of the totals for the line or stub.
The totals therefore are in every case 100% and the bars are of
uniform length — literally a series of 100% bars. The scale for
such a chart is measured off in percentages instead of actual
OCCOyaTlOH*
ib« ««g*-«anilivc popuUiloa
|« «SY£I SOKOPEAK HXtlOXS
ifpl- 0ag»> Itta- Jfaou- C«m»
•ul* portm- ia(> foot* •true*
iwr# «l*l tion •to. wl»e tton
ffiTiTt got eiif^ ■■ EX3 GSSB
taclukd U.r X1.4 *.Z S*0 T.d «.« U.2
Uliivm 21.t U«| 2.0 «.t e.O t.» U.T
«9nuajf SC.l. S.3 2.9 S.S T.O t.O 9.1
tTMM* 41.4 C.S 2.9 l.« 4.4 4.2 12.t
luly w.l 2,4 S.l 0.9 2.1 6.0 U.4
teftrU 90.9 8.9 1.2 I.« 2.9 $.0 T.2
fO.l 2.4 1.6 0.1 2.2 1.6 6.2
Fig, 98. The Relative
Uae- ‘
or Percentage) Bar-chart.
values, and the chart is a ‘^relative^’ chart, or chart of relative
values. Where both absolute and relative charts are being
shown together, it is a good practice to use black for the ab-
solute data and bars and red for the relative data and bars, a
OCCOP^TlOJfS OP THE EMPLOYED
of PoF«l.ticn 10 Y.jjr, Old ^nd Ov.r C»lnful Occupation,
urlt.d States
1920
fSourc* - Dftltod States Census)
Fig. 99. Any Pair of 100% Bars Really Form a Relative Bar-chart.
practice to which the ordinary two-color typewriter ribbon and
carbon papers easily lend themselves.
COMPOSITE BAR-CHARTS
115
A very different kind of chart is the one for which we sug-
gest the name of multiple bar-chart. Here, two or more series
of values, which are in themselves all totals, have been inter-
larded or dovetailed and fitted into each other and the bars
IIT TO SCALES FOB CSURTS
TOTAL
Crud9 for ua« In aanufaeturlns
Foodatuffi In eruda oonditlon and food anianla
Foodatuffa partly et wholly aanufaeturad
llanufaoturaa for furtVMr ua* In aonufactur In^
Manufaoturas raady for eonauaption
Mlaeallanaoua
e,060,4e0,821
l,«ro,767,OS4
917,990,828
1,U«,60S,173
968,496,878
3,204,867,789
11,763,129
laporta (Soala of MilUona of Dollarty
fC®EWH TRADE OT TH* MUTED 8TtTE8
Claailflad nstura of artXolaa
1920
Fig. 100. The Multiple Bar-chart.
should be carefully and clearly distinguished in color and
shading to show to which series they belong. The bars should
generally be made narrower, to fit them closely together, into
little groups, one group for each stub in the data. The data
on the chart can still be prepared in distinct columns, although
PUBLICAIIOM OF TXSH BOOKS
Th* leading nation*
1919 h 1920
(Souroe;- Le Droit d'Auteur, Paria}
(Kote:» Japanea* **book8" not atriotly ooiaparable)
Fig. 101. A Goodi Comparison of Historical Data.
CHARTS AND GRAPHS
ii6
the bars have been interlarded, for the data is more easily
consulted when not interlarded but kept in distinct columns.
It is really a case of bringing two or more sets of bars together
for cross-comparison. The result is rarely entirely satisfac-
tory, and at best is confined to the combination of two sets,
three or four sets being hard to make mutually distinct, and be-
coming confusing to the reader.
In fact, it can be laid down as a general rule that both the
compound and the multiple bar-charts are too elaborate and
complicated. A chart is always better the simpler it is, and we
should make strong efforts to simplify these charts, and if
possible reduce them to simple bar-charts. It usually pays
well for sacrifices we make in this way, in legibility and interest
to the reader, and after all, the chart of this type is generally
directed at a reader, rather than at the maker. The only one
JAPAN
SWITZERLAND
NETHERLANDS
SPAIN
DENMARK
SWEDEN
U. KINGDOM
NORWAY
FRANCE
ITALY
BELGIUM
FINLAND SB
1 PORTUGAL '"H
I GERMANY 8
Fig. 102. Correlation is Indicated by Mirroring.
Ratio of gold reserves of Central Banks to paper currency in circulation compared
with the relation of exchange rates to par value (March, 1922). '^Permission of
Mr. Varl Snyder.
of the three which stands out as absolutely simple and clear
is the relative compound bar-chart, which consists of nothing
more than a series of 100% bars.
The reason for the simplicity of the relative compound bar-
chart is to be found in its uniform length of lines or bars. Only
the segmentation of the bars changes in the chart, and the
reader is not called upon to judge at the same time of various
lengths and parts, but only of the various parts. In order to
secure something of the same symmetry that marks the rela-
tive form, many chart-markers prepare the absolute form of
COMPOSITE BAR-CHARTS
117
compound or segmented bar-chart in a pyramidal or bi-lateral
form, the individual bars being aligned, not with even left'
hand ends, but with even centers, and each bar extending
equally to right and left of the center line down the middle of
1860
SIXTY YEAhS OP IVatlCRATlOM
total Number of ImmigrantB Arrived
United Statee
1660-1920, by years
1,285,349, in 1907.)
Fig. 103. Symmetry has Only a Popular Value.
the chart. Even simple or unsegmented bar-charts are some-
times arranged in this way. The form is certainly more pic-
torial and decorative than the forms which have been described.
But data is, of course, not easily attached to this chart; in
CHARTS AND GRAPHS
fact it is ordinarily omitted. When plenty of space for the
chart is available, and data is not going to be shown, but
extremely pictorial or sensational effects are desired, this
symmetrical form would appear to be entirely permissible.
But it is certainly not to be recommended under any other
conditions.
GERMANY
5CAND1 IRE
NAVIA LAND
GERMAMV
AUSTRIA HUNOm
A CENTURY OP IVyiGHATiOH
Countri** of l«*t p^manont rooidonc* of Immigr*nto Arrlvod
Unitod S-t&tos
1620.1920, by doCAdot
Fig. 104. Connection Lines or Shadings to Distinguish Segments.
A common practice in the elaboration or decoration of bar-
charts both simple and compound, particularly when they
represent connected data, is to draw connecting lines across
“-terc «tt ti
m
<£»
fifr- »*m
*«*4 l«l>or«r«
H«r., .ho, p,ro.„t,(t„, bZ. J' *“**"5
' • '3«*nUti*g)
<r rst
105 . Warping „,
P g the Chart to Show the Tr^ j .
of Chan«.
120
CHARTS AND GRAPHS
observe the effects where the bars are kept and the connecting
lines drawn in. The reader is, of course, enabled to identify
and compare corresponding segments of the bars more easily.
In an extreme form of this chart, sometimes called the “stream
DESTINATIOM OF EXPORTS
EUROPE NO. AMER. S.A.OTHER
Fig. 106. Connection Lines. Note Inserted Data.
Percentage of total imports received from different continents, and percentage
of total exports shipped to different continents . — Permission of Mr. Carl Snyder.
chart,” the bars are broken or bent so that corresponding seg-
ments are kept as close as possible together. The chart is
adapted only for data in which the changes of segments are
fairly uniform and the result is entirely popular in its appeal,
having no research value at all. What will be later described
as a smoothing process, takes place in the segmenting division
lines across the bars, and they, together with their connection
lines, are made as nearly as possible straight rather than
zigzag or rectilinear lines. There is little to be said for either
the symmetrical bar-chart, the stream bar-chart, or the con-
nection lines between the bars, but they are here described as
examples of the modification and variation to which the bar-
chart itself is susceptible.
It would, of course, be possible to go on and still further
combine compound and multiple bars until we have acompound
COMPOSITE BAR-CHARTS
I2I
BOOKS PUBlIShEO IH THE UNITED STATES m EHStAND
Hgw books, new ecfitions and patnphiats pubtished in IScO
compared as to subject, for the two countries#
(Source:* rh« Publte/iers’ l/eekli/, i,
NEW
New
Pam*
Tot**.
800XS
EOITIOHS PHLETS
fleiericsn in
■■■■ C-=3
l^Uin type)
fSritish in
italie tanu)
TOTAl.
6,101
1,086
2,235
8,422
7,676
2,266
3,063
11,004
209
33
32
274
333
SI
13
376
KllttlOl
467
37
161
665
873
77
29
679
SOOIOIOST
353
43
363
769
6«7
61
222
870
70
39
57
166
313
76
74
363
£ovetrtog ''
101
10
123
234
164
17
72
3S3
Ahuoiout
141
54
49
244
276
36
2
30S
Self wee
182
49
281
612
411
94
92
597
TCCMMOtCeY
269
93
163
636
437
155
138
730
MtDIOIMI, HiALtw
132
75
83
290
363
133
64
446
AeaieutTune
49
18
223
290
146
33
39
318
OowesTic EeoNovT
22
6
21
49
57
13
3
72
SuitNtss
144
24
78
246
103
26
19
138
FiNf AXTA
94
6
80
135
158
16
10
184
Music («okk$ about)
44
6
23
72
S3
6
65
Games amo Ssosri
50
10
62
112
120
23
8
161
UITESATUBE
248
S3
SO
351
293
S3
IS
366
PotTsv ANO Osama
409
44
105
558
436
80
47
562
FieriftM
778
345
31
1, 164
1,038
1,051
IS
3, 104
dUviwiLf
410
67
22
499
613
148
0
770
Histosv
503
36
172
711
498
44
43
S3S
Giosrasmv, Travel
144
22
66
222
434
71
109
604
271
14
29
314
840
32
X
374
MtSOILLAWtCUt
21
3
11
36
U1
-
-
181
Fig. 107. A Compound Multiple (Absolute) Bar-chart.
multiple bar-chart. Rarely you may have use for such an
animal, but it really lies out in that field of freak charts into
which the enterprising chartist will inevitably wander by
himself, and from which he will surely return if he keeps his
senses. The field is wide open and there is unlimited oppor-
tunity for originality in the making and dressing up of bar-
charts, but in the last analysis all that matters is to tell a story
and tell it well. You will generally find that this object is best
attained with the simpler, sounder methods which have been
122
CHARTS AND GRAPHS
SEX OP EiaGHABTS AKD IMMIGRAI^TS
Inmigrant Aliens Admitted and Emigrant Aliens Departing
United States
1917-1920
(Source:- Report of United States Commissioner General of Immigraticn)
1917
1918
1919
1920
Fig.
Admitted
Departed
Admitted
Departed
Admitted
Departed
Admitted
Departed
108 , A Compound Multiple (Relative) Bar-chart.
here discussed. In bar-charts, perhaps more than in any other
form of chart-work, we must keep the purposes of simplicity
and clearness always in mind, and avoid the more complex
details which will suggest themselves, insidiously and attrac-
UNITED STATES
perot
51.4
Naw England
T9. 2
Ulddla Atlantic
74.9
Ea»t North Central
60.8
Weet North Central
37.7
South Atlantic
31.0
East South Central
22.4
•West South Central
29.0
Ifountaln
38.4
Pacific
62.4
0RBAH POPULATION
United St&taa
1920
Par cant
Fig. 109 . The Compound Relative is the Best of the Composite
Bar-charts t
COMPOSITE BAR-CHARTS
123
FOREIGN TRACS OF TUB WORLD
Combined Exports k Imports of Leading Nstlona
at par of Exchange
(Source:- United States Statistical Abstract)
Total Trade
Foreign with
(Millions
of Lollara)
World (40 nations)
75.311
14,47G
Lnltad States
(1920)
15,359
13,369
United iCinpdcm
(1920)
15,925
3,123
Canada
(1920)
2,204
1,256
Frarce
(1919)
7,429
i.see
Italy
(1^19)
4,159
1,51b
JJetherlt^nts
(1919)
2,5.29
31fa
Uapan
(1919)
2,421
1,420
Cema:^
(191d)
4,95G
577
(Millions of Dollars)
S W
nnir
■Mi
LTIT;
1 1
Fig. 110. The Simpler Forms Are More Effective.
lively to us as makers and designers. A simple chart which is
read and understood is better than a complicated one which
no one deciphers.
Chapter XIII
PICTORIAL BAR^CHARTS
For purposes of publicity, the circular form of chart has
decided advantages over the rectilinear chart. This has been
gone into in the chapter on pie-charts. The circle attracts the
attention even of casual readers. And circular-shaped charts
are therefore popular with all those propagandists who seek,
by sugar-coating their information, to dispense it to an un-
willing and indifferent public. The charts are useful for, and
should be only designed for, advertising, and the popular pres-
entation of educational matter. They are useless for research
and study. These considerations have been discussed in the
chapter on pie-charts, but arise again in connection with the
possibilities of converting series of bars, that is, bar-charts,
into series of circles.
Truly, when a series of circles are to be used in a chart, the
chart-maker’s road should be marked ^Warning: Dangerous
Curves Ahead.” For the path he must pursue around the
unshakable fact that a circular area has two dimensions, is at
times devious and hard, in view of the rule against showing
one-dimension data by two-dimension charts. The data
which is shown upon bar-charts has but one dimension in the
sense of the rule and the bar-chart itself has but one dimension.
But when the bar-chart is converted into a series of circles,
the result is extremely likely to have two varying dimensions
and be as disastrous as the use of squares discussed in the
chapter on dimensions.
We have seen that the substitution of a single circle for a
single bar (or 100% bar) is harmless, for the reason that the
area of the segments of the circle vary directly with the arcs
and subtending angles. In short, in the pie-chart the areas
of slices of the pie vary directly with the linear or one-dimension
variations, and there is no conflict of measurements. The
ch^art is like the 100% bar, for that too has an area which varies
124
PICTORIAL BAR-CHARTS
125
directly with one of its linear measurements. In the bar the
width is constant; in the circle the radius is constant.
But a series of bars of different lengths can not rightly be
turned into a series of circles of different circumferences. In
a series of bars the widths of the bars can be kept constant
and hence their areas can be made to vary directly with their
lengths. But in a series of circles of various circumferences
the radii cannot be kept constant, but must vary with the
circumferences, so that the areas of the circles will vary by
the squares of the variations of the circumferences. In short,
the moment you use circles of different sizes, the old conflict
between area measurements and linear measurements creeps
in, the most fundamental principle of charting is violated,
and the chart becomes fallacious and deceptive.
There is but one type of bar-chart in which the bars can
be turned into as many circles, and that is the relative com-
pound bar-chart of the last chapter — a chart which is nothing
more than a series of 100% bars. As neither the total length
nor area of these bars varies, they can be safely turned into
as many pie-charts or 100% circles of uniform circumference
and area. The chart is one in which the segments alone are
significant. It is true, as was said in the chapter on pie-
charts, that the segments cannot be so well compared as in
the relative compound bar-chart, and for this reason the chart
is of less value for careful study, but the circular shapes have
been secured and the chart has been perhaps made more at-
tractive and popular. 1
For the simple and multiple bar-charts there is a dodge by
which circles can be used, if you are intent on circles at any
cost. The result is not appreciably less interesting and it has
the advantage of being accurate. It consists in using whole
circles and fractions or fragments of circles, all of uniform
radii. Adopting one value in the series — p.erhaps the average
—as 100%, you must turn all your data into percentages
before preparing this chart and then plot as many circles and
fractions of circles as the data calls for. This method of
charting is sound because throughout the circles and fragments
of circles, a uniformity of radii has been maintained, and the
areas vary directly with the circumferences and arcs.
The drawing of these charts is comparatively easy, as all
circles and parts of circles can be put in with a bow pen or
compass without changing its setting. The work involved is
126
HIGHEST PRICES OF FOOD
lnd«at Humbw# of Hotall Prie«»
United State#
Arerage 1913 • 100
(Source:- Bureau of labor Statiitioa)
*11 article# (Jun, 1920)
Plate beef
(Apr, 1919)
Cbuck roaet
(May, 1919)
Bacon
(dul, 1919)
Lard
(Jul-Aug, 1919)
Cheese
-(Aug, 1919)
Butter
(Deo, 1919)
Coffee
(Jan-Jul, 1920)
Hans
(Apr, 1920)
Bice
(May- Jun, 1920)
Flour
(Jun, 1920)
Potatoes
(Jun, 1920)
Sugar
(Jun, 1920)
Sirloin steaB (Jul« 1920)
Bound steak
(Jttl, 1920)
Bib roast
(Jul, 1920)
Com neal
(Jul, 1920)
Bread
(Jul-Sep, 1920)
ifea
(Jul- Sep, 1920)
Ham
(Aug- Sep, 1920)
pork chops
(Sep-Oet, 1920)
Milk
(Oet-SoT, 1920)
ISERt
(Dee, 1920)
Fig". 111. The Circles Must Have Uniform Radii,
PICTORIAL BAR^CHARTS
12J
far less than it would be if circles of different radii had
been used, requiring fresh setting of the pen for each circle,
SOEEION TPADE OP THE WORLD
Conblned Exports and Imports of Leading Hstlons
at Par of Exchange
(Source;* Onited States Statistical Abstract)
Total Trade
Foreign with
Trade U. S.
(Millions l||fc
of Lollara)
World (40 nation*)
75,311
14,479
(Millions of Dollars)
United State*
(1920)
13,359
13,359
Unlt.d Eingdom
(1920)
15,925
3,123
•••eoxoaxDoooo
Canada
(1920)
2.304
1.256
•©^
Pranoa
(1919)
7,429
1,686
•sooooo^
Italy
(1919)
4,189
1,616
•eoo
Ketharlanda
(1919)
2,639
316
Japan
(1319)
2,421
1,420
•CP
Oantany
(1913)
4,965
577
eooop
Fig.
112.
Segmented
, Like the Compound Bar-chart.
from square root calculations of the variations. The seg-
ments or fragments of circles can ordinarily be drawn in at
TEE mEAL PHILANTHROPIC BUDGET
A Prepoa«4 National Budgot of Phllanthropto Donatloui
Th» Unltod Statoa
1921
Total 11,749,000,000
(SouTC*:- Psnil and DorothT Douglas, "What Can a Man Afford?*)
Fig. 113. Suggesting Metal Coins.
CIL-IRTS JKD GRJPHS
C/D
C/D
w
G
O
a,
O
>-
P4
D
H
w
o
H-1
<
JXI
' o
-j r-,
a
W , ;
t P
*
P ^
w !
P5
o
CD
O
Fig. 114. Pictorial Figures May be Substituted for Bars.
PICTORIAL BAR-CHARTS
129
sight, very accurate work for which protractors would be nec-
essary not being of any value in this chart.
This form of chart is largely an attempt to present bar-
chart information popularly. It is for that purpose particu-
larly adapted to financial data, in which the circles can be
taken to represent dollars and the fractions of circles parts of
dollars. While in strict theory the circles should be at even dis-
tances from each other, yet where there are several for a single
bar or figure, the conception of metal money is so vividly pre-
sented that the circles can be overlapped. The overlapping
or circles saves much space without lessening greatly the im-
pression on the reader’s mind. It is as if, in the West where
silver dollars are still used, you should lay out a row of these
coins, each, except the first, tilted up and resting partly on
the next one. But where space does not require this crowding
up of circles, it is better as a general rule to place them at
AUTOMOBILE PRODUCTION
Number of Passenger Cars Produced
The United States
1913-1S21
(Source:- National Automobile Chamber of Commerce)
Fig. 115. The Third Dimension Is Ornamental.
200
130
CHARTS AND GRAPHS
SAVINGS or trs woBta
rercAplift S&vingB B&nk Oftposlis iri leading na^ione
(Sourcf;» SUtiBilcel Al»«ir«ci)
PICTORIAL BAR-CHARTS
131
even distances. They will then roughly form bars of circles
or coins.
Nor do you need to present a row of circles in the place of
your bars. Rows of human figures, all drawn to the same
scale, can be used in the place of bars. The length of the rows
and the number of figures depicted in each, will show quite
as well as plain bars would have shown, the various amounts
represented. This is a method which those who wish to com-
PKObUCTIOh CP BASIC CCUyCLITItS
Index Pijuros cf Uci\thl> Prcduction in i»pecxfi6d Industrie#
United States
Dee. 1921- Jan. 1922
(Source*- Federal Reserve Bulletin)
(Notaal * Trend after AUoting for Seasonal Variations and y^ar-to»year growth ® 100)K
(Blftclc Arrc# » Jan. 1922)
(’Ahlte Arro# » Dee. 1921)
(Dotted Arrow • Low of 1921)
Fig, 117* Pointers, Instead of Segments, Suggest Pressure Gauge Dials.
132
CHARTS AND GRAPHS
V ^
FURKITURE AUD
293 FURiVlSHIiNGS
^288 CLCTHIJJa
THE HI5H COST OP LIVIKO
Index Figures of Retail Priceji
United States
June, 1920
CSourca:- Monthly Labor Review')
(1913 Average • 100)
Fig. 118 . Aeroplanes, Horse-races, Boat-races, and the Like, Have a
Certain Popular Valuer
PICTORIAL BAR-CHARTS 133
pare pictorially two or more populations, can safely employ.
Instead of showing the Japanese army with a single small
soldier and the American army with a single large one, in
which case you confront the reader with three conflicting
measurements — height, surface area, and cubic volume or
weight — you need merely show one Japanese soldier and sev-
eral American ones, all of the same size, and their number will
give an accurate conception of the relative sizes of the two
armies.
Here, then, is the answer to the problems raised in the
chapter on charting principles. Here is the proper way to
show pictorially the comparison between two or more items.
Do not draw one loaf of bread and an enlarged replica of it
beside it, to show how much the food-bill of the nation has
changed, but draw one loaf of bread and label it with the
earlier year and draw several loaves of the same size and label
them for today. Do not bring together a large and a small
house nor a large and a small nugget of gold, nor a large and a
small railroad car, but place together a single one of each and
a group. The number of times this simple rule is violated,
with results which vary between gross understatement of a per-
fectly good case and gross deception about a poor one, will
amaze you when you begin to watch for it. And the amount
of money spent sometimes in publishing them, futile or false
as they are, will also amaze you. It is one of the most frequent
of all errors in charting.
In addition to the geometric pattern of the rectangle and
circle, there are countless pictorial devices, the simplest of
which is to indicate a third dimension to the bars, setting
them up on end for this purpose. The ingenuity of adver-
tising artists has hardly been tapped as yet, and thermometers,
barometers, or pressure-gauge dials are but the beginning of
the avalanche. The pictures of motor, horse, or boat-races, or
altitude flights of aeroplanes have already been found useful,
and it is probable that all popular contests can be made to
yield attractive pictorial substitutes for the prosaic bar-chart.
Chapter XIV
VERTICAL- BAR CHARTS
Between the sensational picture-bar which we have just
considered, and the plain bar itself, there is a type of bar
which is both accurate and popular. Of less value in the stat-
istical laboratory, it nevertheless deserves a passing glance
even from the most academic investigator, for it forms an
interesting link between bar-charts and higher things.
To make a bar-chart popular, knock it over flat on its side,
so that the bars stand up on end. Simple, isn’t it f But that’s
tl.fi
t-M’
•a#
«.4 »
•.tf
•.Si
».* •
1,0 1
t. s I
•.a t
u. ii
iBOMotiMi XI na mm Mufii’
atttrfkwtlMi «f «rM« fktu
JkMlwtfMia m4nmi0, tMtutlmt
X*a4l»< VtatM, SflS.
Fig. 119.
the rule. There being nothing more to discuss in the matter
of making popular bar-charts, we are tempted to close the dis-
cussion at this point and produce a pleasant surprise to all.
!«•
XftftMM
134
VERTICAL-BAR CHARTS
IJ5
But the vertical bar-chart is rich in suggestions for the higher
forms of charts which we are approaching, and it deserves
a close study.
u:a i»it mi
«l,lll.40«,000 l•^*4•,474,000
90 W mum or m mu)
Fig. 120.
The chief value of the ^^pipe-organ chart’’ as It Is sometimes
called, lies in the realistic picture it gives of quantities. From
a base line these quantities are seen to rise the full length of
the bars, as so much substantial material stacked neatly in
piles where we can compare them. We view them from the
level or floor on which they are piled. We do not have to
climb up and get a bird’s-eye view of them as in the ordinary
bar-chart, where we seem to be looking down upon rows and
rows of goods, but we see them from a natural view-point.
Nor do we rely upon an arbitrary arrangement by which their
left ends have been brought together as in the bar-chart, but
we know instantly that if they are piled up, it is their tops
which we must watch. The pipe-organ chart finds instant
response in our minds, and appeals to us as both logical and
natural. A child can comprehend it.
If you call this base-line the x-^'slxis of your paper and give
the upright bars values in the y-axis, you will be reminded of
co-ordinates and maps. But it is not necessary to go so far.
Merely think of your own back yard, and the nice high fence
about it which you have just white-washed. Assume that
through some weird freak of carpentry you built it with
CHARTS AND GRAPHS
136
IHBAT ilHOB OATS COUTOH OIL CORN AUTOMOBILES
OHIIED STATES l>EBCENTA(}E OP THE WORLD’S PRODUCTION
tS fpeolfied commodities
Fig. 121.
boards which run horizontally. Or turn and look at the wall
of your house, with the weather-boarding running horizontally
about it. Against such a wall let us pile your quantities in
neat columns or let us stand up some dark boards of the right
height against it. You are then ready to take a photograph
which will be a good pipe-organ chart. The lines of the
weather-boarding on the house will make the field of the chart,
and the upright dark boards will be the bars.
Note also, and this is important, that if through standing
too close you should take a picture showing only the upper
ends of the upright boards, but not their full lengths, you
would consider the resulting picture not only a failure but
actually deceptive. In other words, you must not omit the
zero-line or base-line. While you would succeed in showing
the variation of the top ends more clearly you would no longer
have comparable lengths. One board might be but a tenth
longer than the other, but by cutting the lower eight-tenths
VERTICAL^BAR CHARTS 137
out of your picture, it would appear to be twice as long. The
thing simply could not be done, unless you wilfully undertook
to deceive yourself or someone else. The conception of the
pipe-organ chart is sound and fundamental. It is perhaps the
most direct charting method we have. It is almost fool-proof,
which is more than can be said of most charts. And it estab-
lishes clearly the vital principle not to omit a zero-line.
Moreover in the pipe-organ or vertical-bar chart, we first
encounter labelling or data difficulties. And if there is one
motto which we should like to print at the bottom of every
page in bold-face type, as do the publishers of other valuable
reference-books, it is this: ^‘Never separate your chart from
its data.’’ On the contrary, incorporate the data in the chart.
For a chart without its' data is a poor lost thing indeed. And
the unhappy reader wishing to know what it means must hunt
FATAL INDUSTRIAL ACCIDEMT RATS3
for* apecxflod industrioa
Unitad Statas
1913
{S«ur«e:*tr, 8. Bureau of Ubor Stati€tte«» Bulletin W)
Bataa par 1000 wortceri
Fig. 122.
138
CHARTS AND GRAPHS
and hunt and hunt till he locates the particular information
in some distant table. As a matter of fact, he won’t do it,
for before he has found his data he has lost his interest in
the matter, and then what good is your chart ?
In the pipe-organ chart, however, it becomes difficult to
append data directly to the bars. Following your rule of
tipping the horizontal bar-chart on end, you would naturally
have the data down below the bars, reading upward laterally.
This is at once a logical and a sound place, for the bars should
be in line with their own data. But because the vertical bar-
chart is for popular consumption, and because the average
man does not care to crane his neck to one side and read on
edge, objection is often raised to this method of disposing of
the data.
0EATE RAttS U) WKKfAMt
IhktU* «nd IlU«a*« ttoaxh p*T 1000 Scldilart fmr T#*!*
In Specif ltd Warn
l&46.1«ia
(Icwcn.o th* Offiatnl 0nlt«d Stntnn BullaUo)
Fig. 123.
VERTICAI^BAR CHARTS
139
Nevertheless the method remains proper, and one is almost
tempted to say, let the average man learn to crane his neck
if he wants to check up on our plotting. As a matter of fact,
the average reader is generally satisfied to know that the data
is there where he can get it if he wants it, and so does not
bother to look at it anyway, An occasional figure in which he
ACCIDENT MORTALITY
Oealh-Rat«i per 1000 Population of Each Group, and Sex
United States
1910-1912
(Source: • Mortality Statiatlea, United States Ceneue)
Fig. 124.
is really Interested will be read carefully by him in spite of
its reading upward, never fear. And throughout the whole
field of charts it is of such great value to be able to place one’s
data or figures out along projected lines from plotted bars, or
points, that we must adopt the upward reading data in spite
of its temporary strangeness. It is to be accepted and adopted
as a proper feature of charting.
Where the bars are very wide, or the spaces between them
wide, there may, it is true, be room in which to write the data
horizontally, in little boxes below the charts. This method is
wasteful of space and compresses words and figures confusingly.
CHARTS AND GRAPHS
140
^ O «( «] u
< H as £«, a t>
but it Is a “very-simplest” method which you will sometimes
want to use in presenting large and simple diagrams to school-
children. If you have the space to give to it, it is perhaps the
better method for extremely popular work, but it is not to be
VERTICAL-BAR CHARTS
141
generally used as it is not in the long run satisfactory even
for the average popular chart.
Two principles can therefore be garnered from the pipe-
organ chart, first that the base or zero-line should never be
omitted, second that data should be kept with the chart j-g.
S2,4
APR. AU(r. DEC.
, 19
APR.
m2
Fig. 126 . An Absolute Multiple Bar-chart.
Volume of Foreign Financing in the United States and in the United Kingdom,
in millions of dollars (pounds converted at current rates of exchange). — Per^
mission of Mr. Carl Snyder,
gardless of the direction in which that data must be written.
And as you progress further into charts, not only will it help
you to retain these principles, but it will also help you im-
mensely to visualize to yourself again and again this chart
composed of vertical bars, a chart from which most of the
higher forms have been evolved,
142 CHARTS AND GRAPHS
Per e«nt tjf
Vimta
through
dutfide
Contaott
Par oant of
Vftsta
through
Manegamant
Par cant of
Taata
through
Labor
Poroant of
mcata
through
Outaida
Contact*
Parcant of
Waata
dua to
iCanagamant
Parcant of
laata
dua to
labor
llan’a Building
clothing
aanufac*
taring
Printing Boot and
ahoa
manufac-
turing
Uatal Textlla
tradea manufac-
turing
THB BLAMB POH IKDUSTHIAI WASTE
in apacifiad Induatriaa
(Source:- The Elimination of Waata.)
Fig. 127. Wide Bars with Data Inserted^
VERTICAL-BAR CHARTS
143
oth*r
Injuri##
Ukcert'tloM,
tut* tnd
brviM*
N
s
7.9
, ------'1
=».»■ «.i
I ^
Otljfff
in;jurl*t
SuffocAtioit
fr«ctur«»
Sprain* or
dltlco*ticn 0
Burn*
L*c«rt.tlon««
out* and
factor 1*1
Bulldinc tnd Uinlivi; tnd
«ncin*«rlne quarrying
THt MATURE Of IMDUSTRlAt ACCIOQiTS
Mow York St«t*
1911 - 1913
(Souroo:- M, Y, State Dopt. of Lakor}
Fig* 128. Connecting Lines Are Often Usefut
144
CHARTS AND GRAPHS
Gross tonnage of world seagoing iron and steel ships in 1914 and 1921 (in millions
of tons ). — Pernnsiiun oj Mr. Carl Snyder,
Chapter XV
CURVES
It may not have been a very clever fellow who invented
curves, but he was assuredly lazy. For he balked at the task
of drawing vertical bars in the pipe-organ style and he said,
‘^Since I am only interested in the ends of the bars, I will
place a dot where each bar ends, and let it go at that/^ And
later when he wished to find the dots quickly, he drew con-
necting lines between them and, behold, he had a “curve/^ A
curve can, therefore, be defined as a line passing through the
upper ends of the bars in a vertical-bar chart.
1790
3,930,000
leoo
5,310,000
1810
7,240,000'
1820
9,640,000'
1630
12,670,000
1840
17,070,000
1860
23,200,000
I860
31,400,000
1870
36,600,000
1880
60,160,000
1890
62,900,000
1900
76,000,000
1910
92,000,000
1920
106,700,000
THB POPlJLATIOS OF
tBB mllTED stAisa
1790-1920
Fig. 130. Here is the Data — Historical.
Let us step out into your back yard again and take another
look at the upright boards which in the last chapter were left
us
146
CHARTS AND GRAPHS
standing against the wall of the house. Will it not be an ex-
cellent plan — if the house is not yours — to drive a nail into the
wall above each board' to mark its height? Then we can throw
the boards away or let the children play with them. And if
tw pommcw or m aassju staibs
we run a piece of dark string along from nail to nail, we will
not have any difficulty in following the changes in their posi-
tions. Here we have a home-made curve. A photograph of
this piece of string (as long as we also show the ground in the
m vamjxxm w me toiso sxams
iTOO-wao
Fig. 132. Vertical Bars for Popularity.
CURVES
147
picture) will do quite as well as a photograph of the original
boards, for we can always imagine the boards running from
ground to string. The picture will be complete if we run
laths of uniform length up where the boards have been, that
the exact position of the nails along the. wall may be clear
when we come to make the next curve on the same wall. In
a chart these up-and-down laths which serve merely to mark
the horizontal position of the now invisible bars, are called
‘‘ordinates'’ and their distances along the ground from the
first lath are called “abscissae" — ^words never to be forgotten.
You will already have observed the wonderful thing about
ITM i«op ww laao vm im wao imp iho
m fOMunoi or rto: oinnB sum
Fig. 133. A Curve Through the Barg*
TBf pormrio* or tee mnwo
1790-1920
§ § § § § I I § § § § I I I
Ills £ I § I i i i I I i
- - - - a a 3 d s s « s’ 2 S
110-
200-
“1
Z _
70-
«»-
50-
.
"
-j
z
40'.
50,^.
b
!
30-
10-
od
1
Ji
2
a
-
-
-
-
- -
I'/so leoo lexo leso is^o laio isso iboo im isso lasK) i»oo X9io 19S0
Fig. 134, The Bars Disappearing; the **Fieldi*^ Appearing.
148
CHARTS AND GRAPHS
lEB POPtTLAlIOM OF THE CHITSD STATES
1790*1920
Fig, 135. The Evolution of the Curve is Complete.
a curve, namely that it is easily combined with several of its
kind upon a single chart. Multiple curves are far better than
multiple bar charts. A number of curves wiggling across the
page at the tops of invisible bars are eminently more satis-
factory than actual bars interlarded. In the first place, com-
parison of several series of data is greatly facilitated in curves
because each set has been condensed and simplified into a single
line. There is no difficulty in comparing values of each series
with each other. In the second place, such a comparison is
more accurate in curves because all similar points on various
sets or series have been brought together upon a single vertical
line. Had we placed the bars in this way on top of each other,
the longest one would have wiped out or hidden all the shorter
ones and in the multiple bar-chart, therefore, each set of bars
coo'ooi'soT , — , — , — , — i i i i , — , — , ozet
CURl^'Eii
149
has to be shifted slightly out of position to avoid the next set.
But when we plot only the end points of these bars, the short
and the long ones show up equally clearly and can be brought
together upon their true ordinate lines. In the third place a
curve is much more easily drawn than a bar-chart. This is a
most important reason, betv/een ourselves. And, fourthly, to
the reader who understands it (and it is a fact that schoolchil-
dren understand it, however little the present generation of
adults may) the curve is less confusing and more easily read
for its salient points. You will find still other reasons why
curves are advantageous as you go on.
A curve cannot, however, always be used in the place of a
bar-chart, for the line which connects the various points im-
plies that the data itself can be considered connected. Much
data can not be so considered. A careful inspection of the
data will soon show whether it is connected or not, for the
stubs of connected data always form a variable. In the chap-
ter on dimensions and variables, the test for variable nature in
stubs was given somewhat as follows: ^Tan the stubs or items
be shifted up or down in their arrangement freely or is their
order naturally fixed by their nature.?’" Variability is shown
by the rigidity of order.
This limitation of the curve method can be made clear by
two or three illustrations. We have before us the statistics of
the population of the United States for each ten-year period
during the last century. The stubs for each figure of popula-
tion in this case are: 1790, 1800, 1810, 1820, 1830, 1840, and so
on down to 1920. Now no sane person would think of ar-
ranging these normally in any order except from the earliest
to the latest, or from the latest to the earliest — it would be
ridiculous to adopt an arrangement such as the following:
1910, 1810, 1800, 1860, 1920 and so on. Clearly, this is a case
in which the order of the items or stubs is naturally determined
by the data itself and these various years can be considered
as the various values of one variable, namely “time.” The
data can be charted on a curve. Consider another example.
In taking a census of the buildings in a certain well-known
village, the investigators returned reports of the number of
one-story houses, the number of two-story houses, the number
of three-story houses, and so on up to 5 5-story buildings.
Here again the order of the items or stubs, that is the number
of stories, is definitely fixed by the nature of these items and
15 °
CHARTS AND GRAPHS
the number of stories can be considered a variable in the same
way as before and the data can be shown by a curve. Take
IMPOHIS INTO HUSSIA
1921
(Sourc«:* Russian Information and EoTiov^ London)
(♦)
Total
124,281,000
Rood stuff*
16,061,000
initBal productd
39,606,000
Timber end seed
504,000
Earthenware
227,000
fuel, pltoh, etc*
2,786,000
Chemlcale
2,032,000
Ketalfi, oree, machinery, tool*
29,184,000
Paper and peper goods
3,977,000
textiles
15,206,000
^Bearing apparel, stationery, etc#
13,132,000
MlseelXaneoua (inoluding 48,000
1,667,000
tone of *• famine aid**)
Fig:- 136. Here is Data not in Series.
another example. The United States exported to England in
the year 1920 a large amount of copper, wheat, rubber, auto-
mobile supplies, machinery and paper. If we were charting
these exports, it makes no difference whether we show the
cotton exports before the paper exports or vice versa. The
order of these items is not fixed by their nature and can be
arranged in any way we desire. Here, then, is a case in which
we cannot use a curve but must fall back upon the bar-chart.
In short, while the bar-chart can be used for all data, the
curve-chart can only be used for data of which the stubs form
values or readings of a mathematician’s variable.^
^ A curve or connected line in the place of vertical bars> for abstract or geograph-
ical data (that is» data of which the kubs are not an ordered numerical series) is a
granhic monstrosity, fortunately not often seen.
CURVES
151
IMPORTS INTO RUSSIA
mi
{SoWOOi* Ru 0 fi&n Infomatxon and Raviair, LondMl)
m
toUl
124,281,000
<
Foodstuff*
16,061,000
Animal product*
39,605,000
Timber and seed
504,000
Earthenware
227,000
Fuel, pitch, etc.
2,786,000
Chemical*
2,032,000
Metal, ore*, maohinary, tool*
29,184,000 1
Paper and paper good*
5,977,000 1
Taitlla*
16,206,000 1
Wearing apparel, atatioiiery, etc.
13,132,000 1
MLlfoellaneoua (including fimin*
1,667,000 1
aid 48,000 ton*)
Fig. 137.
No Curve
(lailiona of Bollart)
I I I I 1 r 1 I I y.] I ■■ j..TTT
If we examine the field or background of a curve, we will
find that it is drawn up according to the principles of Cartesian
co-ordinates which we have already observed. The reader
who has forgotten or omitted that weird chapter had best
turn back to it and read it carefully. In order to understand
co-ordinates for curve-chart work, you must know that the
x-zxis of your chart is that straight horizontal line along the
bottom of the chart which we sometimes call the base line of
the chart, or zero line. The y-axis is the vertical line whose
value at all points is zero on the ;t-axis. In the ordinary chart,
and y axes are along the edges of the chart, the A;-axis at the
bottom, and the y-axis at the left hand side. In this case
there is no room on the chart for negative values. It is not at
all uncommon, however, to have charts which reach over and
beyond the two axes for the plotting of negative values. In
still other charts, the true y-axis does not appear at all, the
chart not showing no zero x-value whatsoever. This is gener-
ally a case of data in which zero itself is meaningless or arbit-
rary. A common example of it is historical data, such as the
first illustration in the last paragraph, where the time values
commenced with the year 1790 — ^not with the year zero. The
horizontal lines, and particularly the distances along the
x-axis, are called abscissae. The vertical lines crossing the
ends of the abscissae or points on the ;?;-axis, are called ordin-
152
CHARTS AND GRAPHS
ates. All points having the same abscissae or values along
the :JC-axis lie in the same ordinate or vertical line, and vice
versa, all points having the same ordinates or values along
the y-axis lie in the same horizontal line.
As we have seen, the curve chart requires data with two
dimensions, the curve being plotted upon a field which has
two dimensions. Along the horizontal dimension or A;~axis you
will find the values of the independent or ^-variable, generally
the stubs in your table of data. For each value of this variable,
that is for each stub in your table, there is a corresponding
value of the dependent or y-variable, namely the figure in the
column beside the stub in your table of data. This y-value is
plotted along the ordinate or vertical line from the given point
on the :v-axis (or abscissae, indicted by the stub) to the height
upon the y-axis indicated by its value. Another way of express-
ing this is as follows: In the data for a curve-chart each
figure to be plotted has two values, one being the value of the
figure itself and the other being the value of its stub in the
table. These two values of a figure describe the co-ordinates
of the point by which the figure is plotted on the chart.
Not only can the point be plotted from the data showing
its co-ordinates, but the process can be reversed and the co-
ordinates of a point can be read from the plot or chart, merely
by following the intersecting lines through the point to their
respective axes. For it will be seen that every point on the
paper has two co-ordinates, one of which is the abscissa or
horizontal line passing through it, and the other of which is
* the ordinate or vertical line passing through it. The point
itself is sometimes called the ^'intersect’’ of these two lines.
‘ As two perpendicular lines can intersect at one point and one
point only, there can be only one point described hj any two
co-ordinates. We can therefore locate or identify any point
by its co-ordinates and the co-ordinates of a point may be
said to fix rigidly its position.
If our data tell us that a certain town has a population of
3,000 persons in the year 1910, we should plot this population
by moving along the horizontal or x-axis, to the distance or
abscissa of the year 1910, and then moving upward along the
ordinate or vertical line through that point, to the height of
the horizontal line (or abscissa) passing through the point of
population, 3,000, in the y-axis. The dot, or point on the paper
which would indicate this town, would be placed at the inter-
CURVES
153
section of the ordinate for 1910 and the abscissa for 3,000,
and the co-ordinates of that point would be "^^ar 1910, popu-
lation 3,000,’’ that is 1910, y, 3,000.” And if we see the
point of this town plotted upon a chart, we can read from its
co-ordinates the information that in the year 1910 its popula-
tion was 3,000, simply by following the two co-ordinates of
the point out to the axes of the chart.
The distinction between the dependent and the independent
variables is important. The independent variable is normally
formed by the stubs in the tabulation and the dependent
variable by the corresponding figures, that is, the figures in
adjoining columns. The tabulation is more or less optional
however, and for certain purposes stubs and data may be
interchanged. The distinction between dependent and inde-
pendent variables goes deeper, and finds its origin in the
peculiar nature of the data. When the readings along any
variable are made a basis of classification of data, then that
variable is the independent one. In general, the dependent
variable is that one whose values may be said to depend upon
the values of the other variable. Such dependence need not
take the form of a mathematical equation or explicit function,
but is merely a matter of convenience in such matters as the
classification and arrangement in the statistical table. Our
chief concern is with the plotting of the data upon curves, and
the important rule to be remembered is that the independent
variable should be laid off on the x-axis and the dependent
one on the y-axis. The rule is not without its exceptions, but
these should always be founded upon special considerations
and in the absence of such special reasons, the rule should be
invariably followed.
Chapter XVI
FIELDS
Most of the good things in this world involve some Sacrifice-
Curves are no exception. In a curve the direct visible connec-
tion between the curve itself and the zero line, or A-axis, is
sacrificed. As time goes on and you become more and more
used to the curve chart, you will begin to think of its values
laW 1870 1880 1890 1900 1910 i8«0 1870 1880 1890 1900 1910 iSCO 1870 1800 1890 1900 1910
1880 ' 1870 1880 1890 1900 1910 *
«0
10 I [ \ I I J
1860 1870 1860 '1890 1900 l«lOi
From Mr. John WenzeVs '^Graphic Charts that MisUadr in Scientific American Supplement,
June 6, 1917. ,
Fig:. 138, The Amputated Chart is Deceptive.
154
FIELDS
1 55
as In some mysterious manner floating disembodied along the
connecting line which forms the curve. You will be tempted
to forget that the quantities rest very substantially upon the
floor (base line, zero line, x-zxis or whatever you want to call
it), and that it is only their tops which reach the points plotted
in the curve. And forgetting this, you will try to save space
by omitting the zero line and lower part of the chart, and by
showing only that small portion or band of the chart through
which the plotted curve travels.
This practice of omitting the zero line is all too common,
but it is not for that reason excusable. The amputated chart
is a deceptive one, tempting the average reader to compare
the heights of points on the curve from the false bottom of the
amputated chart-field, rather than from the true zero line, far
Ullllons §
1917 ^ **GrapMe Charts that Mislead'* in Scientific American Supplement,
Fig. 139. The Case Against Amputation is Clear*
CHARTS AND GRAPHS
156
below and invisible. A curve-chart without a zero line is in
general no whit less of a printed lie, than a vertical bar-chart
in which the lower part of the bars themselves are cut away.
The representation of comparative sizes has been distorted
and the fluctuations (changes in value) exaggerated. In a few
more years, the principle that the zero line, when zero is real,
must normally be shown in a curve will be universally accept-
ed. Then the emphasis which now must be laid upon this
principle, will not be needed. Indeed, the author plans in his
fortieth edition of this work, to omit almost all reference to the
rule. But, today, you will repeatedly find violations of the
rule complacently propagating false impressions. And today
the principle must be iterated, reiterated, and forever kept in
minT
CURATIVE EPFRCT 0? DIPHTRERlA ANTITOXIH
Temperature record of typical caee of Diphtherie with prompt uee of Antltoxiii
iSouroo;- U. S. Public Health Service)
Uaye:- 1 2 3 4 6 « T
Fig. 140. When Zero is Arbitrary, it can be Omitted.
There is but one case when the omission of the zero-line
on the y-axis or dependent variable, is justified. This is the
case in which zero itself is an arbitrary value, and does not
really mean a “nothing.” As we have seen in the case of
FIELDS
^57
;c;-values m historical data, the year 0 does not really signify
zero years, but merely signifies an arbitrary point of time
from which counting is begun. Science has many such arbi-
trary zeros; in the Fahrenheit scale of temperature, for in-
stance, zero degrees is really an arbitrary point. Common
sense will tell you when the zero is a starting point of the
quantities measured by your data. And whenever zero is
really such a lower limit, the rule that the zero line must be
shown on a chart applies.
Sometimes, even with the best of intentions, rules must be
violated and we must do the unjustifiable. The usual excuse
for amputating a chart is that to show the zero line would
require too much space, or would reduce the scale and make
the fluctuations of the curve less noticeable. Sometimes you
will feel the force of this argument very strongly. It is par-
ticularly frequent in charts of dividend, interest, and yield
rates, where the fluctuations are in percentages and the base
is understood by everyone to be 100%. The argument has
greater force when the chart is intended chiefly for circulation
among those in a profession who are already accustomed to
think of the minor variations of percentages, and who would
study the chart with interest only with regard to its time to
time fluctuation-quantities, but would have no interest in its
relative total quantities. Here it may be argued that the am-
putated chart would deceive no one and would be of greater
service than if, at the cost of detail, it were made complete.
When this argument arises, it must be scrutinized with
care and hostile scepticism. Often the argument will be found
specious, resting more on the familiarity of the chart-maker
himself with his data than upon the true attitude of those
who will see the chart. In other words the maker of the chart,
in the thoroughness of his own understanding of the data,
forgets that others will be less familiar with it, and attributes to
them his own skill and comprehension. It is easy, in this
way, to be modest about one’s own powers of understanding,
but the modesty is costly when, as so often happens, the reader
is loath to admit his inferiority, and merely lays the charts
aside for study at some time "later on” which time, needless
to say, never comes.
Only if it is quite certain that no misunderstanding will
result, should the chart be amputated for the sake of saving
space or exaggerating fluctuations. And even in such cases,
CHARTS AND GRAPHS
158
great care should be taken to make the amputation self-evident
to the most casual reader of the chart, for it is precisely the
man who has little time for study of the chart, who is most
likely to be deceived by it. The best method of making the
amputation of the chart obvious is to blot out with Chinese
WORKERS OUTPUT ASD FATIODS
lnde:)c Nuabdrt of Hourly Output in Doxterou# Handwork Operations
United States
(Souroe:- United States Public Health Service, Bulletin Ho. 106)
Couaiutator 66.2 97*2 97,2 100 95.5 94.3 93.2 SG*5
Magneto tapi ng 9*2.4 100 95.7 95.1 96.8 96.6 96.0 91.5
Roll coll 96.1 98.8 100 98.4 92.2 98.4 98.8 88,3
Rivet press 91.2 96.9 100 94.6 94.5 92.9 94,2 89.1
Arerage 91.2 98.0 98.2 97.0 94.8 95.6 95.6 88.9
OOOOu>»0«OiO«« o
tQ l0,»O «r-« r-t r-«
«> 00 OJ* O « r-* W W*
First Second Third Ipourth'^ Fifth Sixth Seventh Eighth
Fig. 141. The White Zone Warns the Reader.
white a small irregular zone across the lower part of the chart-
field, or to erase the co-ordinates in this zone, and show the
zero-line below this zone. The chart then has the appearance
of being broken off between the zero-line and the curve, and
anyone will see that it does not show full distances to the
curve-points. An easier but less effective method is to make
Fig. 143. A Wavy Base-line is a Shorthand Warning.
Adjusted Index of the Volume of Manufactu re (100= Normal).
^ The use of rounded or dotted base-lines to indicate abbreviation, which is some-
times advocated, seems ill-advised, since the method is not self-explanatory and hence
defeats its own purpose. The object is to flash to the casual glance the abbreviated
condition of the chart, and any symbolism which must be technically understood is
of no more value than the scale-figures themselves for this purpose.
CHJRTS AND GRAPHS
160
the reader than the true zero-line, from which he should meas-
ure the quantities shown by the curve, lies far below the visible
portion of the chart.
Another principle which will quickly appeal to your common
sense, is the rule that when zero is real, the zero-line should be
extra heavy to make it prominent. Remember that it takes
the place of the floor or lower end of the bars in the bar-chart.
It should stand out, therefore, in such a way that the reader
can easily grasp its significance and compare with it the
heights of the points on the curve. The rule is particularly
important in cases where the chart extends down below the zero
line into the negative side in order to show negative and posi-
tive values. On the same principle the 100% line, when it
occurs in a chart, should be similarly heavy as it also may be
IKC01£E C? RAILROADS
Ket R&llTfflLy Operiiting Incos,»6 of Cl&«a I Rends
thcBo having wnr4ual Operating Revenues in Excess of $1,000*000)
Ur.itea States
1920-1921
(Source - Interstate Coaaerce Coaaaission)
(Note:- Set railway operating inccne is total operating revenue lese
total operati.ng expense, railiray tax accruals, uncollectible railirey
revenues, equipment and joint facility rente.)
FIELDS
i6t
considered a base for zero points, being the point of zero loss
or gain. In fact, the rule may be extended to all cases of lines
showing significant constant values, and the zero line should
not be heavy, unless it has a special significance. In charts
showing temperature in Fahrenheit degrees, for example, since
the zero point is merely an arbitrary value like any other
number of degrees, it would be more sensible to emphasize
the freezing and boiling point lines. Common sense must be
relied on to determine the lines which can be usefully em-
phasized.
The ordinary curve-chart has three different types of
figures which must be attached to it. These figures are: first,
the scale figures for the A;-axis, showing the values assigned to
the vertical lines; second, the scale figures for the y-axis, show-
ing the values which have been assigned to the horizontal lines;
and third, the data figures, showing the values represented by
the various plotted points of the curve. There is considerable
confusion and difference of practice in the positioning of these
three sets of figures. By going back to the first principles,
however, and recognizing that a chart is merely a fragment of
the co-ordinate system of measurement, we can easily find the
logical and natural places for these figures, and it so happens
that the positions which are the most logical have proved in
practise the soundest and most useful ones.
The scale-figures along the horizontal or .r-axis are really
the values of the independent variable. In your table of data
they form the stubs or items. As you read in the last chapter,
this independent variable belongs on the ;^-axis; it should never
be placed along the y-axis, as is sometimes erroneously done.
The proper place for the scale showing these figures or values
of the independent variable, is at the bottom of the chart,
each figure or value being immediately beneath the lower end
of the vertical line to which it has been assigned. Do not,
merely for the sake of ornamentation or decoration, place
these values at the top of the chart also. Do not box them in,
each with a little square or circle. Do not make the printing
unnecessarily large. Do nothing more than is necessary for
simple clear results. These precepts will save you a great deal
of time in the preparation of your charts and will save your
reader much trouble in its reading. It is enough to place the
figures once for all at the bottom of the chart, forming a scale
along its entire base.
CHARTS AND GRAPHS
1 6a
The figures which are assigned to the various horizontal
lines should be placed in a column immediately beside the
ends of these lines. They form the scale for the dependent
variable or y-axis. In the case of isolated charts, that is charts
which will appear singly and alone, this vertical scale is often
placed at both sides of the charts so that the reader can read
the values of the horizontal line at either side. For isolated
charts, there is no particular objection to this practise, though
it may be a work of super-erogation. But, in the majority
of cases your charts appear in groups, and often on separate
sheets which the reader will wish to place side by side for the
purpose of comparison. Then certainly the two vertical scales
would be a nuisance; one is sufficient, and it should be placed
at the left hand side. Indeed, it would be best to make this
rule universal, namely, that the vertical scale should appear
PHODOCTICM OF ACTOyCBlLES
flu&ber or pasaetiRer Cara ard Trucks Producarf
Unitad States
1913-1521
(Source;- Satlonal AutcmoMle Chaebor of Cpiaaerce)
rruok* I i I i I I § I I
^ ^ ^ ^ ^ ^ ^
Paaeenrer
i §
§ §
Fig. 145. The Sound Position for Two Vertical Scales.
FIELDS
163
once only, and then at the left hand side of each chart. The
use of two scales, one on each side of the chart, is desirable
only for more popular results.
When several curves are shown upon the same chart, it is
often desirable to use different scales for them. That is, the
same horizontal lines may be given two or even more different
values for different curves. But even in these cases, it is better
to place both scales, once and for all, at the left hand side.
The practise of placing one of these scales at the right hand
side, and another at the left hand side, has little to recommend
it. Theoretically, at least, the left hand end of your chart is
normally the y-axis itself, and the scale or scales should logic-
ally be attached immediately thereto. In practice this logical
position is justified.
rCRCtHT.
Fig. 146. An Interesting Comparison of Different Period*.
Here it is not the y-zxls, but the .^-axis, which has two scales.
Monthly price index of 14 basic commodities during two war periods. Pre-war
year in each case is taken as the base of 100%. Prices of the same commodities
are included for each period . — Permission of Mr, Carl Snyder,
We come then to the question of the third set of figures,
namely, to the data itself, which the curve represents. As we
have said before, this data should always be presented with
164
CHARTS AND GRAPHS
the chart.2 It should not be omitted entirely or separated and
printed in an appended table. That rule is one of the most
important in chart work, and he who violates it fails to afford
his reader with convincing proof of the accuracy of his chart.
We must find a way to insert the data in the chart, and typo-
graphical difficulties or inconveniences must not be allowed to
deter us.
As a matter of fact, the place for the data is obvious. It
should be placed at the top of the chart, each figure immedi-
ately above the point by which it is represented on the curve.
The reader can then glance down the ordinate or vertical line
through any particular point and find the stub or value of the
independent variable, and he can glance up the same ordinate
or vertical line, and find the value of the curve at that point,
that is, the exact value of the dependent variable. Needless
to say, he could have found this latter, the dependent variable
value, with approximate precision, by careful study of the
horizontal line through this point and its intersection with
the y-axis.
In entering figures on the chart for the scales of the inde-
pendent variable and for the data itself, we come to the same
typographical difficulties which we met in the pipe-organ or
vertical bar-chart. We do not often find sufficient room to
print or write these figures horizontally on the page. Even if
the chart is so large that we can, by fine printing, crowd the
figures together horizontally, we will generally find the results
unsatisfactory. In the first place the figures tend to run into
each other, and in the second place they lie across, rather than
in line with the ordinates or vertical lines to which they are
attached. If we attempt to box the figures in with squares,
circles, or diamonds, we merely add to the confusion of the
chart and detract from its simplicity.
2 It is obviously the chart from which data has been omitted, which has led Professor
Secrist to say:
^‘Tabulation of classification precedes; the use of diagrams follows. The
former geneially serves to clarify the meaning of data; the latter frequently
to obscure it . . . Diagrams alone are more likely to serve as bases for
conclusions arrived at without study and to foster a disregard for the details
from which diagrams are drawn , . . Diagrammatic illustrations can
never leplace data themselves, no matter how accurately they tell the truth
or how illuminating they are. They are at best statistical aids and should be
so viewed by those who use and study rliem. A well-drawn and cleverly
executed diagiam is never a guarantee of the value of the statistical facts
which it illustrates.** — Secrist, Horace, Jn Introduction to Statistical
Methods, The Macmillan Company, New York, 1917, pp. 159, 161.
FIELDS
165
The sound principle, therefore, and one which will be found,
after a little practise, eminently satisfactory, is to enter the
data-figures and all except the simplest ;?^-scale figures, vertic-
ally, that is, by writing on edge. The reader has little diffi-
culty in turning the page about to read these figures when he
wishes. There is, therefore, no great disadvantage to this
method. The figures lie clearly along the lines of the ordinates
to which they belong, so that there is no doubt or confusion
in finding the figure for any particular point on the curve.
You will notice, moreover, that the figures for the independent
variable and the figures for the data arrange themselves in
the familiar form on your original tabulation, the only differ-
ence being that a chart has been inserted sideways between
the stubs and data. And this is logically sound, because the
curve is merely a modified form of the pipe-organ chart, and
the pipe-organ itself is merely a bar-chart placed on its side.
Even the column headings are retained in the curve chart in
their same relative positions to each column of figures.
You will find it useful to keep this relation between the
curve-chart, the pipe-organ or vertical bar-chart, and the bar-
chart proper or horizontal bar-chart, always in mind. Par-
ticularly so, when you have several columns of data to be shown
by several curves upon the same chart, for in this case it is
important to retain the column headings at the top of each
column of figures in the data. These column headings will
then be to the left of the chart itself and the only diflterence
will be that they can be written on horizontal rather than
vertical lines, so that they can be read easily while the curve-
chart is in its normal position, though the data is on edge.
You will, however, find it useful to re-arrange the order of the
columns of data so that the position of each corresponds
roughly to the position of its particular curve, the data for the
uppermost curve being at the top, and the data for the lowest
curve being at the bottom of the series of data on the chart.
In order to distinguish two curves on the same chart, which
may or not cross each other, you will probably use different
colors for these curves. In that case, it is useful to observe a
similar color distinction in the printing or typing of the data
columns, each column being printed in the color in which its
particular cuiwe is plotted. You will also find it useful to place
a small sample section of this curve immediately beside or
underneath the column heading for the data to which it is
CHARTS AND GRAPHS
1 66
attached, thus forming a sort of key to the curves used on the
chart.
For those who use typewriters (and in all large offices, it is
well to use typewriting exclusively for the lettering and figure-
writing on charts) the arrangement above described for the
positioning of figures and data, will be found extremely con-
venient. It gives the typist no more trouble than the prepara-
tion of an ordinary table or tabulation of figures. Placing the
chart sideways in the typewriter, she types in at the left hand
edge of the chart-field the stubs or items of the table, and at
the right hand end of the chart, the figures of the various
columns. If the ordinates of the chart have been arranged at
the precise typewriter distances of 3 , or | inch apart, she
has no further adjustment of the paper to make in the type-
writing machine. In line after line down the page, she merely
reproduces the table of the original data leaving a wide gap
between stub and columns of data — a gap which is filled up
by the field of the chart. The chart can then be plotted in
upon this field after the data has been typed.
The whole process of making a curve chart takes no more
time than that of making a bar-chart, and in fact, very little
more than that of making a plain mathematical table. The
only instructions the typist must have are the data (which can
be in the form of the original table) and clear orders as to ( 1 )
which columns must be copied on the chart, ( 2 ) the order in
which they must be placed, and (3) the color, if two colors
are used, in which they must be typed. The only instructions
which the draftsman needs are then contained in the form
itself on which he is to draw, his instructions being the data
as already typed on the chart. If the vertical-scale figures for
the y-axis of the chart have also been typed in, his instructions
are complete. But unless the chart belongs to a standardized
set in which the scale has been fixed, the draftsman will prob-
ably determine upon his y-axis scale after a study of the data
itself. In this case, he enters the scale in hand-lettering and
proceeds with the plotting. If this scale also is to be type-
written, however, he would enter the scale-figures only in
pencil and the typist would enter the figures permanently
last of all. The considerations affecting the choice of scale
will be found in the following chapter.
Chapter XVII
SCALES
Technicians are fond of describing a scale in puzzling and
abstruse language. Yet, as often happens, the thing itself is
so simple that a child can understand it. It is usually defined
as a ratio — the ratio between actual distances in the space
charted and equivalent distances on the chart. This ratio of
reduction or enlargement is important to engineers but not
to the maker of mathematical charts. We therefore use the
word scale for the ‘calibrations measuring distances on the
chart. To linear distances, both horizontal and vertical, we
assign arbitrary values and the figures which tell us these
assigned values form the scale.
Curve charts take up two dimensions on the paper, that is,
they have both a vertical and a horizontal axis, and therefore
require two scales. These two scales may or may not be alike.
When they are alike, we have what might be called a normal
projection. Imagine a simple chart, the field of which is square
and the two scales of which are alike. Draw a straight line
from the point of origin or lower left hand corner of the chart-
CIURTS JND GRJPIIS
1 68
field at an angle of 45° to the horizontal and extend it diagon-
ally across the field to the upper right hand corner. This line
passes through all points having equal co-ordinates, that is
equal values along both axes. Now let us see w'hat happens
to this line when one or the other scale of the chart is changed.
Suppose we shorten the vertical scale to half of its distance.
Relatively speaking, this is the same thing as doubling the
horizontal scale. (By half or double the length of the scale,
we mean assigning the measurement values to distances half
or double as great.) Now the result of this will be very
noticeable upon the slope of the straight diagonal line passing
Fig. 148.
through points having the same values as before, for the line
will rise only half as much as before. If our line were a curve
wiggling across the chart, its wiggles would be half way
Fig. 149.
flattened out, giving us the impression of much less fluctuation
than formerly. But as a matter of fact it is exactly the same
curve as before, only its field has been changed so as to dim-
inish the vertical oscillations or wiggles.
SCALES
169
Fig. 150 .
On the other hand, suppose that we increase the vertical
scale to twice the length of the horizontal scale. This is the
same thing, relatively, as reducing the horizontal scale to half-
size. Now see what happens to the diagonal line. Its slope
becomes far steeper than originally, as it must climb to twice
the height in the same horizontal distance. If that line had
been a curve, snaking its way across the paper, its wiggles
170
CHARTS AND GRAPHS
would have been twice as great as formerly. It would have
given us the impression of a very unsteady and changeable
proposition indeed. First way up, and then way down. Very
hard to tell just where it is going to go next. Unstable, un-
reliable, fickle — these are the conclusions we should have
formed of the items that were charted, and yet those items are
precisely the same as appeared on the second chart above
described where their movements appeared to be very even
and regular.
In short, the scales on which a curve is drawn can affect
very much our impressions of the data by magnifying or minim-
izing the apparent movements of the curve itself. Of course,
this does not mean that the relative height from the base-line
of the various points on the curve have been altered. If you
have been careful to show the base-line always, the base-line
SCALES
171
itself will approach nearer to the curve as the vertical scale is
reduced and the wiggles are flattened out, and will recede
Fig. 153.
farther from the curve as the vertical scale Is enlarged and the
wiggles are exaggerated. But it means that the oscillation or
fluctuation of the curve will have been made to appear more
violent or milder according as either of the scales is changed.
And it therefore behooves us to give serious thought to the
matter of scales before we determine upon them finally for
any particular chart. As a matter of fact, we may have to
try out several combinations of scales before we find one which
gives just the right amount of emphasis to curve fluctuations
to suit us.
S ' • Now where our chart size is unlimited, and we are free to
extend the scale and field in either direction as far as we wish,
this rule of try, try, try again might be perfectly feasible.
172
CHJRTS JXD GRJPHS
Perhaps, in that case, we would generally come' back to the
normal projection or combination of two similar scales. If,
as often happens, the scales measure different and incomparable
(technically “incommensurable”) quantities, such as years on
one axis and dollars of sales on the other, or length in inches on
one axis and weight in pounds on the other, we could change
these to percentages (each of its own total or maximum), and
consider the percentages commensurable.
But generally, the space available for a chart is limited.
If it is to appear in a book, the size of the book-page must be
conformed to. If it is one of a set of charts, a uniform chart-
size increases the attractiveness, if not also the simplicity,
both of the set and of the individual chart. Even if the chart
is to appear entirely alone, there is much benefit in avoiding
unhandy sizes. Moreover, worrying through a succession of
trials consumes time and energy, a needless waste if it is true
that we can determine beforehand merely from the data
itself what will be a satisfactory combination of scales.
Let us consider the horizontal scale first. In the previous
chapter we have already found certain considerations which
will affect the arrangement of the figures for this scale. These
figures will be placed Immediately below the base-line or bot-
tom of the chart. Normally, they will be written on edge, up-
ward, or typewritten after the paper has been fed into the
typewriting machine sideways. The ordinates or vertical lines
to which the figures belong will then, if extended down the
chart, pass through the figures, cutting across the middle of
each digit. In the same way, above the top of the chart the
data-figures will be placed, each on line with its own scale-
figure and plotting point and each so placed as to be similarly
cut by the extension of its ordinate.
It takes no brains to see, therefore, that the horizontal
scale must be large enough to permit entering the figures, no
matter how condensed, of the data. As a general rule, type-
writer intervals, which are in picas or sixths of an inch, are
about as small as your horizontal unit-distances should be.i
And if you have a short series of data, you can double or
treble this distance without expanding your chart too much.
In fact, the curve is more easily read when the horizontal
^All typewriters can be especially equipped, at a slight extra cost, with any
desired interlinear distance and there is one machine, the Hammond, frequently used
in academic work, which has intervals of one-ninth instead of one-sixth of an inch.
SCALES
173
units are about typewriter double-spacing distance apart, that
is, three to the inch.
We are assuming here that your finished chart is to occupy
a sheet of paper about standard letter-size, 8^ by 11 inches. If
larger sheets are to be used, you will modify all dimensions ac-
cordingly. Where charts are to be exhibited in a large room
to a large audience, they must be many times larger and all
oS
ix4
U
0}
u
a.
Fig. 154. Examples of Convenient Horizontal Scales.
Facsimile Typewriting.
lines and lettering correspondingly heavier. The most excel-
lent chart in the world is virtually useless to the man who
cannot see it, and you must not forget the distance from
which the chart is to be viewed. For ordinary study, however,
as well as for convenience in handling and in filing, the 8^^ by
11 basis is satisfactory. It can always be enlarged by photo-
174
CHARTS AND GRAPHS
stats, or photographed on lantern slides for very large pro-
jection. (,
The range of this horizontal scale then depends largely on
the number of items in the series, to be plotted. A series
•which contains more than thirty items had best be cut up into
Fig. 155. Showing One Month by Days on Letter-size Paper*
Single-spaced for typewriting data.
two charts, each one of which will run across the shorter dis-
tance of sheets of the 83 ^ by 11 paper. You can generally do
this by breaking up the series into convenient segments. Thus
if the data is monthly, break it up into years and present a
year on a page. If the data is annual, break it up into ten
or twenty year groups. Where it seems inadvisable to break
up the series into parts this way, double width sheets can be
SCALES
ITS
used, either folding up into regular size or not, as desired. If
you wish to run the chart along the long distance of the 8^4 by
11 paper, the space for attaching data above the chart will be
much restricted, but if the data is limited to one or two col-
umns, this is no disadvantage, and you can get as many as
Fig, 156, One Year by Months on Letter-size Paper,
Double-spaced typewriting.
fifty-two items on a page, thus enabling you to show a year
by weeks.
When you can do so, it is always well to make the hori-
zontal distances one-third inch each, or double typewriter-
spaced. In this case, you cannot count on more than a dozen
or fifteen units crosswise on the paper and twenty-five length-
wise. The advantage of the wider spacing, as has been said,
CHJRTS AND GRAPHS
176
lies in the greater ease with which it is read, neither its ciir\e
oscillations nor its data figures being so confusingly close to-
gether as when smaller spacings are used. Moreover, the chart
with fewer items on it will generally be more closely studied by
the reader than one with a great mass of detail In fact it some-
Fig. 157. One Decade by Years.
Triple-space J typewriting.
times pays to omit minor details in the data and make the
items fewer and more important, in order to reduce a great
amount of detail to a simple series. Thus the daily stock quo-
tations would require a very large chart for their presentation
during a year, while the weekly and sometimes the monthly
average quotations will be just as significant, and far simpler,
to the reader.
SCJLES 177
Now as to the vertical scale.2 The first general rule is that
the highest plotted points on the curve should ordinarily reach
about two-thirds of the way up the field of the chart. This
gives the best results, because the top of the chart neither
Fig, 158. One Quarter-century by Years
Single-spaced typewriting.
crowds the curve too closely not does the space above the
curve seem to the reader unnecessarily large. If the top of the
chart is too close to the curve at any point, the reader may be
^ It is to be understood in the following discussion that what applies to the posi-
tioning of a single scale for a single curve applies also to two or more scales for two
or more curves when these are shown on one chart. Unless there is special reason for
having one curve below the other, or for using a common scale for both curves, the
second curve may have its own scale (lettered on the chart beside the first scale)
specially positioned, like the first scale, to bring the second curve to similar heights
upon the chart.
8 CHARTS AND GRAPHS
SCALES
179
led to measure with his eye distances on the chart from the
curve to the top line, instead of from the bottom line.
When a single long series of items is to be carried through
many charts, one after the other, forming a set in which the
individual charts show only parts of the series of data, it is
important to have the vertical (as well as the horizontal) scales
uniform throughout. The uniform scales are necessary that
Fig. 160 . One Year by Weeks on Letter-size Paper.
Single-spaced typewriting.
the charts may be individually compared, or “fanned out^^ into
one long series of continuous charts. And if the scale be such as
to place the highest point in the whole series three-quarters of
the way up the page, in one chart, there may be other charts
in the series, in which the curve will hardly leave the zero, or
base-line. This cannot be helped without enlarging the scale
for these smaller parts,, and so destroying the comparability of
the charts. There is no help for the low charts in this case,
nor is help really desirable, since the lowness of the curve at
certain points is the significant fact to be shown.
The size of the vertical scale depends therefore upon the
amount of the largest figure in the date. We must glance
SCALES
i8i
through the columns of data to be charted and observe the
highest quantity in the series. Of course if this is a freak
quantity, we can disregard it and select the next highest quan-
tity (leaving the highest one to extend clear out of the chart
if it will). Having determined on what we shall consider the
high point or “peak” of the data, let us substitute for this a
round figure, which we shall position about two-thirds or
three-quarters of the way up from the bottom of the chart.
PIRE LOSSES
Cnitod StcLtas
1876-1920
(Souroa:- Journal of CoBniaoroa)
Thus if the high-point or peak is 38,370,000, let us take38,000,-
000 as the scale-making figure. Now if this peak is approached
by several others, that is, is no unusual value, let us make it
slightly lower down on the chart, but if it is unusual, and the
gperal level is nearer six or four million, let us make it slightly
higher up. Anything between a third and a quarter of the
distance below the top of the chart is sufficient. Assuming
that the series contains several figures near eight million, we will
select the slightly lower position, and place eight million two-
i 82
CHARTS AND GRAPHS
thirds of the way from the bottom of the chart to its top. In
other words the entire vertical distance will be divided into
twelfths, each representing one million dollars. In this way
our vertical scale has been determined.
Of course, the size of the chart itself has not yet been
settled. Its width we disposed of under the head of horizontal
scales. But so far we have not settled its total height. We
have only decided the number of parts into which that total
jMi 5,902,000
Peb 5,640,000
tor 6,413,000
Apr 6,494,000
toy 6,309,000
Jun 6,186,000
Jul 6,329,000
Au« 6,261,000
top 6,360,000
Oct 6,619,000
Her 6,147,000
Dec 8,370,000
Dhited Cigar Stores Co. 8al«i
1921
(Dollar*)
(Source:- Survey of Currant Bminaai)
Fig. 163.
height will be divided. But the total height need give us
little trouble. For a drawing on ordinary letter-size paper,
the chart field, that is, the co-ordinate rulings, should not
cover much more than half the height of the sheet of paper.
There will be some space needed at the bottom for the hori-
zontal scale figures, and considerable space should be left at
the top for the data and for the title to the chart. So on a
sheet of paper 11 inches high, the chart can best be made
about six inches high. And in the example we have just con-
sidered, where this total height of six inches will be divided
into twelve parts, each representing a million dollars, it is
easy to see that the ordinates or horizontal lines should be
spaced half an inch apart. The scale ratio is f inch to 31 >000,-
000 .
SCALES
183
There are many devices for dividing a length into desired
divisions. The case of ten, fifteen, or eighteen or more divi-
sions is no more difficult in a six-inch space than that of
twelve divisions. Both engineers’ and architects’ rules divide
the inch into various useful numbers of parts and when we
desire an odd or fractional number of parts per inch, we can
United Cigar Stores Co. Sale*
1921
(Dollars)
CSource*' Survey of Current Businea*)
W3IACOU>(0<0(0<0<0<0«>CO
easily get them by drawing parallels from the corresponding
divisions on a regular scale laid off so as to form a triangle
with the desired scale and the last parallel. However, as we
had considerable margin of choice in deciding the number of
dividions, we can always find round figures which will work
easily in the given chart field.
The fact is that a standard “field” about 4 inches wide and
6 inches high, h^s already been adopted by a great many chart-
184
CHARTS AND GRAPHS
Fig. 165# Commercial Forms Available.
Four Useful Charting Papers published by Mr. John Wenzel, Yonkers, N. Y. The first three are of
type described in the text to fit the 10 or SO, 30 or 60, and 20 or 40 sides of the Engineer’s rule.
The fourth is specially adapted to percentage data, requiring a scale from 0 — 100 %^
SCALES
185
makers, and by some publishers of chart-paper. This standard
prepared chart-form is very useful. It is printed low upon
regular letter-size 8^-by-l 1-inch sheet, leaving the necessary
space below the chart for horizontal scale figures and a great
deal of space above the chart for data and title. It is generally
printed with the horizontal rulings only, so that any desired
number of vertical rulings can be drawn in to suit your hori-
Fig. 166 . To Obtain a Scale Smaller Than Those Given by the Ruler.
zontal scale. And the horizontal rulings are printed without
any scale-figures for the y-axis, so that you can adopt whatever
vertical scale you please. Three different rulings are made, in
which the interval is either a fourth, a fifth, or a sixth of an
inch. These three forms are sufficiently different to enable
you to place a point almost anywhere you wish on the field,
merely by selecting the right ruling and attaching to it the
proper calibrations or scale-figures.s
2 A single chart-form, which has intervals of one inch up the paper between hori-
zontals, can be conveniently used in place of the three, when only a few charts au^
to be made. It is a master-form which can, if desired, easily be converted into any
of the three by ruling in the proper number of intermediate horizontals.
186
CHARTS AND GRAPHS
Moreover, if you are doing much plotting, it will help you
greatly to use what is called an “engineer’s scale” ruler, which
can be obtained with the inch divided into fourths, fifths, and
Courtesy of Kenffel Esser, N. Y.
Fig. 167. Engineers^ Triangular Rule.
sixths, coinciding with the three types of prepared chart-
paper, and into other even fractions, namely halves, thirds, and
tenths of an inch. With these six scales suitable for use with
the three types of ruled paper, you can conveniently draw up
any curve in any position on the field. And if you have not
access to the specially ruled paper and rulers, you can of
course easily prepare them for yourself.
For this chart field which measures 6 inches high (or for
any other given size of chart) a general instructions-table can
be used by which, without any figuring, you will know what
Fig. 168 . To Obtain a Scale Larger Than Those Given by the Ruler.
SCALES
187
type of ruled paper, what edge of the- ruler, and what values
or calibrations in the chart-scale you must use. In the ac-
companying table for 6-inch high chart-fields, it is assumed
that you will position the peak of the curve about two-thi/ds
of the way up the chart-field. You have therefore only to
glance through your data and find out the amount of the peak
or largest quantity in the series and with this figure in mind,
consult the table and find the round figure therein which is
nearest it, and proceed as for that round figure. The only
w 60 *0 80 eo 4& 60 60 40 40
Figo 169. Examples of Convenient Vertical Scales.
Reduced from Standard 6-inch Field.
Shows scales with 10, 12, 16, 20, 24, 32, 40, 48, 60, 80 and 100 at the distance of
two-thirds of the height of the six-inch chart-field, by the use of the three rulings
for 40, 50, and 60 sides of the ruler.
thing to remember is that the position of the decimal point
does not matter. Your number may be .0003 or .3 or 3.0 or
30 or 3,000,000; you will always find its plotting instructions
under the first ^^significant’^ digit, namely, in these cases, the
figure 3,
These apparently arbitrary rules of thumb are justified
only so long as they serve to produce the best results. Your
real purpose is to show the data most clearly and simply,
either to yourself or to someone else. The chart is a window,
i88
CM J RTS AND GRAPHS
TABLE OP SCALES POH CEASTS
Fig. 170. Table for Vertical Scales with Engineer’s Rules on 6, 8, and
10 Inch Fields.
as it were, through which the reader looks out upon an illu-
minating picture of the facts he is considering. Through this
window he sees, if you like, a chain of mountains, whose
height tells him the values or quantities he is considering.
That he may see them to the best advantage, the window must
be low enough for him to see the base of the mountain-range
SCALES
189
and high enough for him to see at least some sky above the
highest peak. In general, the best view of the mountains
would show neither too much nor too little clear sky above.
And if the window is crossed with a framework for small
window-panes, he can further judge of heights by the criss-
cross window-pane lines. Your curve is the silhouette of that
mountain-range, your field the tiny window-pane outlines, and
you, the chart-maker, must use your own judgment and ar-
tistic sense to place the reader’s chair near or far, high or
low, in front of that window, to give him the clearest view.
Chapter XVIII
PLOTTING-POINTS
A point can be defined or located by its co-ordinates. The
co-ordinates of a point are the two mutually perpendicular
lines which pass through it and at whose intersection the point
is located. One of these lines is its abscissa, the other its or-
dinate. Neither of them need appear upon the paper; that is,
they may both be imaginary, and it is therefore sometimes
difficult to chart or plot a point precisely, or when plotted, to
read its co-ordinates exactly.
Fig. 171.
Here is a simple example. Suppose you have a chart-
field on which each abscissa and ordinate represent a unit
value, that is the abscissa and ordinate are numbered consecu-
PLOTTING^POINTS
T9T
tively one, two, three, four and so forth. Now suppose that
you were to plot on the field the point represented by the co-
ordinates, 3K, y, You will look in vain for an
intersection of lines with these values, because the co-ordinates
of half units have not been drawn on the paper. Nevertheless,
you can imagine the two co-ordinates, one of them half way
between the ordinates of 3’’ and AP the other half way
between the abscissae of and y, 5.’’ And at the inter-
section of these two imaginary co-ordinates you can plot the
point.
This question of the precise plotting-point comes up very
often in charts showing time by weeks, months, or years along
the horizontal scale. Suppose you are charting the monthly
steel prices in 1920. Down at the bottom of the chart you
will place the time-scale with the words January, February,
March, and so on under the ends of the vertical lines of the
chart. Consider this scale carefully. What does it mean ?
It means that each unit of horizontal distance has been taken
to represent one month and that all twelve horizontal units
tiaken together represent a year, the year 1920. Now if a month
were a single instant of time, it would be very simple. | We
would then plot the figures for each month or single instant of
time on the particular vertical line which represented it. But
as a matter of fact, a month is a long period of time with a
great many diflFerent instants in it, all of which go to make up
a single month, just as twelve months taken continuously
make up a single year. In short, we are no longer dealing
with single instants of time but with continuous periods of
time. Yet on our chart the horizontal scale shows the number
of single points representing these months. Something surely
is wrong. Obviously we must find particular instants for
points of time, to correspond with the points on the horizontal
scale representing time.
There are two ways of doing this. The more scientific and
accurate way is for us to seize upon the particular instant or
point of time between months and represent these points of
time by the points on our horizontal scale. The origin of the
x-axis, or zero point on the horizontal scale will then stand for
the beginning of the month of January. The first point on
the horizontal scale will indicate the end of the month of
January and the beginning of the month of February. The
second point on the scale will represent the end of the month
192 CIURTS JXD GRJPHS
of February and the beginning of the month of March, and so
on. In this case we see that the months themselves are indi-
Fig. 172. To Plot Anywhere Between Ordinates.
cated on the scale, not by points, but by spaces between
points. Thus if we wish to plot the figures for January as of
the 15th of January, that is the middle of January, we will
find a point midway between the zero and first upright lines,
that is in the middle of the first space on the horizontal line.
To plot a figure as of the end of the first week in January we
will locate the point only a quarter of the way from the begin-
ning of the horizontal scale to the first point on the scale, that
is, a quarter of the way from the first of January to the end of
January. And the scale itself, that is, the words ^^January,’’
'^hebruary,"’ ''March,’' and so on, must be placed beneath the
various spaces between lines, and not beneath the ends of the
vertical lines themselves. This method enables us to distin-
guish prices at the various parts of each month, and is in general
the more accurate method of scale calibration and point
plotting.
The other method, however, is more convenient both for
chart-maker and chart-reader. Let us assume that each month
PLOTTING-POINTS
1:93
has been condensed into a single instant of time and that the
upright line or point on the horizontal scale represents only
Fig. 173. To Plot Only upon Ordinates.
this single instant of time. What particular point of time in
the month is in general the fairest one for us to choose to
represent the whole month? Obviously the middle of the
month or the middle of the fifteenth day. And when we plot
the prices for January we assume that those prices are the
average prices for the entire month and that it is fair to show
them as the prices for this point in the month, namely the
fifteenth of the month. In this case our scale figures will be
written immediately underneath the ends of the vertical lines,
that is, the word ^^January"^ will appear under the first ordin-
ate and not under the first open space, the word ^Tebruary’’
will appear under the second ordinate, and so on. Clearly
this method is not scientifically so accurate as the first method.
But as it is much more convenient, it is the ordinary method
of plotting a time series. When you use it you must remember
the assumption upon which it is based, namely, that the entire
period has been condensed into a single moment or instant of
time and shown as of that moment. Whether that single
194
CHARTS AND GRAPHS
moment be at the middle of the period or at some other time
during the period will depend upon your data, and somewhere
about the chart a memorandum should be placed showing
what particular moment in the period shown, the figures and
plotted points represent.
This second method can safely be used whenever the periods
for which the data are charted, are uniform and equal periods
of time. The method becomes very difficult and confusing
FOOD PRICES IM PRANCE AND OREaT BRITAIN
Index Humbert of Retail Food Prices xn Franco, Great Britain, and United States
X920
(July 1914 • 100)
(Sourca:- Bureau of Labor Statistics)
Franco (For current
(excluding Parle) Quarter)
o>
•o
<t> o
OO MO
Groat Britain (For current
two Months)
o
uO
CM
O
«>
w
Fig. 174. Data with Different Intervals.
however, as soon as the time interval of the data changes.
Suppose that a part of the year was represented by monthly
PLOTTING^POINTS
195
average figures and part of it by weekly average figures, and
perhaps also a part by quarterly average figures. You would
have to watch your step in plotting these various periods by
the second method. But by the first method it is all smooth
sailing, for it is easy to plot diflFerent points in the space when
the spaces represent the months. It is also easy to plot a
single point in the middle of three successive spaces (as for a
quarterly period) by the first method. The first method, as
has been said, is sound and logical and should be used whenever
you are in doubt as to the plotting point on the scale. This
means also that it should be used whenever the time intervals
are not uniform and regular.
It is not necessary to say that one of the reasons why points
should be correctly plotted is that the reader of the chart
should be able to ascertain the values they represent directly
from their co-ordinates on the chart. And the reader may
have particular need for these values, not at the points plotted
from the data,' but at other points along the connecting lines
which form the curve. The technical name for the process of
locating points on a curve between given ordinates is ^^inter-
polation.” Just as we can interpolate for points on the scale
when plotting given points, so also we can reverse the process
and interpolate for the data of points upon a given curve. In
the foregoing we have shown how to interpolate foj plotting
points in making the charts. Let us now consider the reverse
process of interpolating for data, either in the making or in
the reading of a chart. It is to this process that the term
interpolation is ordinarily applied.
Let us suppose that we have a chart compiled from data
which is incomplete, that is to say, the months of June, July
and August are missing. Our chart will show a curve extending
over the first five months and the last four months of the year
but there will be a gap during the three missing months. Yet
we know that the phenomenon was in existence during these
months. Therefore we cannot leave this gap vacant, for to
do so would imply that the phenomenon had ceased to exist
during that time. Now if we can make no guesses whatever
as to the value which was missing, we should simply draw a
dotted straight line across the gap connecting the two nearest
known points of the curve — dotted or broken to indicate that
data is missing and the curve for that distance is guess-work.
If however, we can make a shrewd estimate as to the shape of
196
CHARTS AND GRAPHS
the curve across the gap, perhaps from a study of the same
phenomenon in other years during the same months or from a
study of similar phenomena during the same year, then we
will not draw a straight line across the gap but will shape the
cur ve over the gap in that way which we think most likely to
be true. It would still be a good plan to use a dotted or broken
line for these estimated or interpolated months. In any event,
we have now assigned values to these months which were
missing, by interpolation from a study of the surrounding
ones which were known. The values of the interpolated points
for these missing months can be read off from the chart and
would form estimates for them with which we can even fill
out the data record.
trade DNIOS MHfflEKSHIP 0? IHE WORLD
Umber of monibers in 20 Comtries
1910-1919
(Source;- Interaatlonal Labor Office)
Huniber
oi’
Meuibera
CM
to
O
o
o
o
o
o
<o
to
o
o
o
o
o
o
<>
lO
o
o
o
o
o
o
45.000. 000
40.000. 000
35.000. 000
30.000. 000
26^000,000
20,000,000
15.000. 000
10.000. 000
5,000,000
0
.
j
\
1
1
rv>
0>Cr><S>0iO0J0i0>0>0>0^^^
Fig. 175. Interpolation for the Period of the War and Extrapolation
for the Years after 191$.
Interpolation for intermediate data from a completely
known curve is also frequent. Thus if a curve shows the values
at certain known points, we can easily secure the values at
other in-between points by noting the points passed through
PLOTTING^POINTS
I97
by the curve, that is, by interpolating for them. This guessing
or estimating process can also be carried out beyond the limit
of the curve to points lying outside of the range of the known
data. A frequent example of this is the well-known process of
Oil Consumption
800
700
I 600
500
cQ
400
I ^^00
i
100
0
1911 1912 1913 1914 1915 1916 1917 1918 I9J9T980 1925 1930
From Joseph B. Pogue's ''Economics of Petroleum,"
Fig. 176. Extrapolation.
Here it is the linear trend (having a mathematical formula) that is projected
into the future.
extending a curve into the future to predict or forecast what
will happen at a given time to come. The curve is simply
projected to points outside of rather than inside of the range
of its data. This latter process is often called extrapolation.
The processes of interpolation and extrapolation are capable
of very general application.
1
'^D
M
ome
silC
1 1
Cons
\umpHon
-r— C-
j
1 ‘
1
Chapter XIX
COMPOSITE CURVES
With the plain single-curve-chart, the reader of this book
is now supposed to be thoroughly familiar. And as has been
repeatedly indicated, the multiple curve-chart is formed by
merely bringing together upon a single chart two or more
curves, which may or may not cross each other. When two
WCDUCTICN OF AUTOUCBILES
Nush»r 6f p*aser.i 5 <‘r C»r» and Truck# Produe#tl
United Statas
191S-1S21
(Sourc* - Rational AutoBoblU Ch*ab«T of Coeaerce)
.JPrue^#^ i § § i
<SA
§ I I
xa o
»- o»
Passer rer
3 S
Fig. 177. Each Curve Has Its Own Vertical Scale*
198
COMPOSITE CURVES
199
curves cross at small angles, it is the better practice to dis-
tinguish them clearly, either by the use of different colors, or
by the use of dotted or broken lines for one and full lines for
the other. An older but slightly more confusing practice is to
adorn one curve with small circles at its plotted points, another
with small crosses, and distinguish other curves with double
lines, wavy lines, lines with small cross-lines and sometimes
lines of different thicknesses. In general, the best results are
now secured by smooth lines, either colored or black, full or
broken or dotted, but all of about equal thickness and visi-
bility. Too many curves upon a chart are far worse than too
INVENTION AND WAR
Nunxbor of patents issued during the Civil War end during the World War
United States
1650-70 and 1903-20.
(Source: - U* S, Statistical Abstract)
World
war
World
war S S S
Fig. 178. Each Curve Has Its Own Horizontal Scale.
200
CHARTS AND GRAPHS
few, and except for special laboratory work a chart should not
normally carry more than three or four curves. The value of
attaching data is so great that it is unwise to dispense with
data, and yet if too many curves are used, the data will bulk
up disportionately. That a few curves can be easily distin-
guished without recourse to special adornments or thin and
PER cent.
Fig. 179.
Number of persons employed in industrial establishments in New York State
and in the United States (figures for December 1914 — 100 %). — Permission of
Mr. Carl Snyder.
thick lines, is obvious. The chief use for extra heavy or wide
lines in curve-making should be for emphasis, as in the case
of one curve for the average or total of the other curves.
A thorough knowledge of the curve-chart, however, requires
at least a passing acquaintance with its sisters and its cousins
and its aunts. We shall therefore hold a reception and intro-
duce the most important of these. Beforehand, however, let
us whisper a word in your ear about them. They are, none of
them, such all around good fellows as the plain curve-chart.
They are not so flexible and universal in their uses. Each
answers excellently to certain limited types of data. We shall
try to make you acquainted with the particular style of data
for which each is best suited, as we meet them.
COMPOSITE CURVES
201
Consider the case of daily stock quotations on the exchange.
For any particular commodity, a dozen dilFerent prices may be
quoted in the same day. It is therefore customary to quote
“highs” and “lows” as well as opening and closing quotations
202
CHARTS AND GRAPHS
in the stock market record. Now if we plot upon the ordinate
for each day both the high and low quotations, and connect
the low quotation points to make a curve for low quotations
and similarly connect the high quotation points so as to make
a curve for high quotations, we shall have two curves illustrat-
ing the same phenomenon, namely stock prices. The two
curves show merely the extreme fluctuations of this phe-
nomenon and the reader of the chart must understand that
prices have ranged between these two curves. To make this
situation obvious, let us shade the area between the two curves
so as to make a zone. The shading can be done either with
gray, with colors, with cross-hatched lines or with solid black
or white, the co-ordinates being wiped out in the last case.
This device conveys at once to the reader of the chart the idea
that prices were not set at one figure alone, but varied con-
siderably within the same day or period of time. The device
can be used for any case of data covering maxima and minima
Fig:. 181 . An Excellent Form of Zone-Curve.
High, low, and average interest rates on commercial paper each year from 1831
to 1920 . — Permission of Mr. Carl Snyder.
'204
CHARTS AND GRAPHS
for a single phenomenon. Climatic conditions, such as
humidity at morning and night, or temperature at mid-day,
midnight, and noon, or tidal variations, or business statistics
such as the margins of profit from individual sales, and in
general all data having a considerable range of variation for
one and the same thing, at approximately one and the same
time, can be shown by this method. This type of curve-chart
is commonly called the zone-curve.
In a sense, the zone-curve is merely a short-cut for a large
number of curves superimposed upon each other. In some
cases you will have such distinct data that you could have
prepared a large number of separate curves. * If you put all
these curves together upon a single chart, you will produce
much the same visual result, so far as the reader is concerned,
as you produce by means of the zone-curve, in which you
merely plot maxima and minima and shade the space between
them. Where the individual curves must be kept distinct
■
■■1
■
■
mi
'H
■
■
■I
■
■
■
■ii
B
■
■
■
■
■
■
li
ii
m
.1/11
1
■
■
mifiii
mill
■
■
■I
m
■
m
IHII
in
m
m
mil
■if
m
■
m
■
■
m
■I
V
■
■II
■■
p
m
■
■
■
■
m
■1
■
■
■
■11
■
■
■
H
■1
m
■
■
■1
■
HH
■III
■■
|H
'HU
■ii
RQ
Bl
IBi
ill
WHM
■
yn
Bi
—
mnim
—1
msm
Permission of Standard Statistics Co.
Fig. 183. An Excellent Adaptation of the Zone-Curve.
however, and compared with each other, of course the zone-
curve is of no use. The zone-curve, therefore, is not a sub-
stitute for the multiple-curve chart.
We have said that the connecting lines between the plotted
points which form a curve, imply a connection between the
items of the data. We have said that this connection is es-
tablished by the variable nature of the stubs, or A;-axis scale.
It is now time to let you into the secret that these plotted
points do not need to be connected. The ‘^gun-shot’’ chart is
an example of a curve without curves. It consists entirely of
plotted points. It is useful for cases of data secured by sep-
arate and often contradictory observations. Each observation
COMPOSITE CURVES
20 S
is plotted as a point but no connecting line can be drawn to
other points and there is merely a large group of dots or plotted
points extending across the chart and showing the result of
various observations. This chart and its data differ from the
zone-curve and its data in that there is no maximum or mini-
mum known. Isolated points or dots on the chart may occur
far outside of the general run or trend or zone of the main
body of observations, such cases being due to freaks, errors,
or other causes.
Gun-shot charts are essentially a research device. They
are often intermediate steps between the first data gathered
RPrt
eOt 002 01)3 005007 oof 02 0
RPn RP«
3 OS 007 10 2 3 5 7 lOO
j ■ - Li- ■ _
■_i ' A ' L-' ''■'i
. I ' ' ' » ■ i i
c.iC.i ... C2 j 3 3 1 '00
Kw Kw Kw
Rj»H RPrt RPI1
Front Leonard A. Doggeit’s "Cost -per pound of Electrical Machinery," in the Electrical World,
Oct. 2, 1915.
Figr. 184# A Gun-shot Chart.
and the final data reported. If we find after making a gun-
shot chart of any particular observation, that all the points
lie within a very narrow zone across the chart, we will be
tempted to draw a line through this zone and so “fit a curve”
to the plotted points. This is a sort of deductive reasoning
by which we may often reduce a mass of data to a simple curve.
A popular form of the curve chart is the “staircase” curve,
sometimes erroneously called a “histogram.” The staircase
curve is a direct throwback to the pipe-organ or vertical-bar
2o6
CHARTS AND GRAPHS
MAOmNfi ADVERTISING
Rumbar of Agate Lines of Adrortising
in Leading Magazines
United States
1913-1921
(Source;- Printers' Ink)
Fig. I8S. The Staircase Curve is Almost a Bar-chart.
chart. If you will I'ccall the original definition of a curve given
in a previous chapter, you will remember that a curve may be
defined as a line connecting the upper end of the bars in a
vertical-bar chart. Now if you will make the bars wide enough
so that they actually touch each other and will then draw the
outline or silhouette of the upper ends, you will have a curve
made of rectilinear lines always parallel to one or the other
axes of the chart. This is the staircase curve. Whereas the
ordinary curve represents the end of the bars by mere points
and connects these points with straight lines, the staircase
curve gives full value to the entire width of the bar. It is
the precise silhouette formed by the bar-chart when the bars
COMPOSITE CURP'ES
207
Fig. 186 . An Absolute Compound Pipe-Organ Bar-chart or an
Absolute Stair-cased Band-chart.
Thousands of soldiers in the American Expeditionary Force on the first of each
month . — Permission of Mr. Leonard Ayres.
are packed close enough to come in contact with each other.
As compared with it, an ordinary curve, directly connecting
the midpoints of the ends of the bars, is called a smoothed
curve.i
In some cases, the staircase chart is more accurate than
the smoothed curve and its representation of areas lying be-
tween the base line and the curve, more accurate. A little
study will show you that the connected-line curve has cut oflF
little triangles from every bar whenever the curve descended
1 Beside the staircased (or rectilinear) and the smoothed (or Hne-and-angle) curves,
there is still a third which is of such doubtful value and great hazard as not to be
mentioned here. It is the rounded curve, in which no straight-lines or angles occur,
but all parts of the curve are rounded oflF by means of “French curves*' or by free-
hand drawing. It is discussed in the chapter on Frequency Curves.
2o8
CHARTS AND GRAPHS
and added little triangles to every bar when the curve as-
cended. These little triangles are sufficient to change the
MAGAZINE ADVERTISING
Nttmber of Agate Lines of Advertising
In Leading Magazines
United States
1913-1921
(Source:- Printers* Ink)
CO
8 § 8
o o
•V* o o*
o o o
cT 00 CM
0> CO I-H
30,000,000
26,000,000-
20 , 000,000
15,000,000
10 , 000,000
S, 000, 000
CTi
lO t- <0^ o> O
rH sH CM CM
O) O) 0> 0> 0> (7>
Fig. 187. The Smoothed and Staircase Curves Differ in Outline and
Areas.
area lying between the curve and the base line, and bounded
by the two ordinates about the plotted point, and when it is
important that this area should be accurately shown, you can
not use an ordinary curve but must use a staircase curve. At
other times the staircase curve is less accurate than the
smoothed one, for its abrupt changes of level give an impression
of abrupt fluctuations in the phenomenon charted, which may
be wholly unwarranted. The considerations governing the
COMPOSITE CURVES
209
comparative value of the smoothed and staircase form of curve
are treated fully in a later chapter.^
The staircase curve is a popular form because it conveys
at once to the average reader the impression of actual quan-
tities between the base line and the curve. Readers who are
confused by ordinary curves find less difficulty in under-
standing this chart. It is not, however, so useful as the ordi-
nary curve because a number of these staircase curves cannot
be satisfactorily put together upon a single chart. Their
vertical portions will so often coincide that it is hard to dis-
tinguish them. The most that can be accomplished in the way
of combining staircase curves is to put two or three of them
together and use dotted, broken, and full lines to distinguish
those which intersect.^
There is a certain type of data for which the plain ordinary
curve closely imitates the staircase curve in its rectangular out-
line. This is the case of data in which the values remain ab-
*6
5
h
3
2
1
1916 1917 I9I8 1919 1920 1921 1922
Fig. 188 . Pseudo-Staircased Curve.
Note that so long as wages remain unchanged the curve must be a straight hori-
zontal line and that when wages change the curve must be a straight vertical line.
Hence the rectilinear form though truly a smoothed curve . — Permission of Mr.
Leonard Ayres.
solutely fixed over a given period, and change only suddenly
and abruptly. An example of this type of data would be the
retail price of a single commodity, which after remaining at
® Cf. Chapter on Frequency Curves.
» Cf. Fig. 293, p. 333.
aio
CHARTS AND GRAPHS
seventy-five cents for a long period of time suddenly and on a
single day jumps up to one dollar, to remain there for another
long period. Obviously to plot the 7Sj?5"Value by a dot at the
beginning of that period and the $1. 00-value by another dot
at the time of change and connect the two by a direct line
would give the impression of a gradual change extending over
RATE RATE
RATE RATE
Fig. 189.
Open market interest rates at New York compared with the discount rates of
the Federal Reserve Bank of New York. Open market rates shown are for
prime 4 to 6 months commercial paper, prime 90-day banker’s acceptances,
certificates maturing in 4 to 6 months, and an average of the yields of 4 issues of
Liberty Bonds and Victory Notes most frequently offered as security for advances.
— Permission of Mr. Carl Snyder.
the entire period. It is therefore necessary that this curve
should be perfectly level until the change takes place and then
jump up to the higher level and remain there. The curve will
then have a rectangular outline similar to that of the stair-
case curve, but the length of time or the length of the curve at
any particular level is not regular and fixed. It is merely an
accident that this picture has resulted in a curve with recti-
linear outlines. It is not the same as the staircase chart.
COMPOSITE CURFES
21 1
RATE
Fig. 190. A Pseudo-Staircased Curve.
Call loan renewal rate and prevailing rate on prime 90-day banker’s acceptances
at New York . — Permission of Mr. Carl Snyder.
BOiro SALES IK BtJKDREDS OF MILUONS OP DOLLARS
DURING FIRST HALF OP EACH TEAR SINCE 1899
Towre oroBS hatched are thoae of hatinesa depretaion.
Note Inoreaaed bond aalea aftoi^ each auoh period.
Permission of Mr. Leonard Ayres.
Fig, 191. An Interesting Use of Shadings in a Band Charts or Vertical
Bar-Chart.
212
CHARTS AND GRAPHS
A gay and giddy member of the chart family is the ^‘band-
chart.’^ Take up any of the ordinary curves which you have
made and with a soft pencil shade the entire area under the
curve. This vividly reminds the reader that the data is rep-
resented by the distance between the base line and the curve
and not by the distance above the curve to the top of the
chart, for it draws his attention forcibly to the lower part of
the chart lying under the curve. You will remember the
THE FAMILY BUDGET
Divided as to Classes of Commodities
United States
1914-1921
(Figures as of December each year)
(Source;- Monthly Labor hevievj’)
Total
103.0
105.1
118,3
142.4
174.4
199.3
200.4
174,3
Miscellaneous
21,9
22,9
24.x
29.9
35.1
40.4
44.3
44.1
Furniture and
Furnishings
5,3
5.6
6.S
7.7
10.8
13.4
14.5
11. X
Fuel end Light
5.3
5.3
5,?
6.S
7.8
8.3
10.4
9,6
Housing
13,4
13.6
13.7
13,4
14.6
16.8
20.2
21.6
Clothing
16.8
17.4
19.9
24.8
34.1
44.6
43.9
30.6
Food
40.1
40.1
48.1
59.9
71.4
75.1
68.0
57.3
Fig. 192. The Curves are True Only for Cumulations of the Layers.
literal representation of quantities used in the bar-chart. And
in fact the band-chart showing quantities by its shaded area
COMPOSITE CURVES
213
can be made in stepping form like the staircase chart as well
as smoothed like the ordinary chart. The staircase band-chart
is therefore even more of a throwback to the vertical-bar
chart, or pipe-organ chart, than the staircase curve itself, be-
cause it has retained the shaded areas of the bars.
The band-chart becomes interesting when it is broken up
into several bands running together across the page, each band
representing a component part of the total amount under the
curve. This is the band-chart proper, a series of layers or
bands going across the chart which, when taken together,
form a total whose fluctuations are shown by the curve of
the top edge of the top band. This chart is sensational and
interesting but of little precise value. You will find it hard,
for example, to measure the width of any band except the
lowermost. In fact the various curves which mark ojBF these
bands one from another have no value except that the lowest
curve is a curve of one segment, the second curve is the curve
for the total of the first two segments, the third is the total
for the first three segments, and so on up to the top curve
which rs the total for all segments or the whole phenomenon.
rREKCH WOMEK^WOKmi. DDRINO THE WAR
proportion of Weaen to Total iii^loyoea la iraaa©
1914-1920
(Sotireo:- Monthly Labor Koriow}
^ ^ s ss
CO O M
o> eo 00
90
80
70
40
30
30
10
0
mm
^
worn
_
S ^ g
-j
I I
i 3
CM O*
a s
Fig. 193* A Relative (or Percentage) Band-chart.
CHARTS AND GRAPHS
a.14
And you will find that area conceptions are inevitably in-
volved in this chart; the reader tries to measure the value of
the various segments by the width of their bands. And
unless staircased these areas will be extremely deceptive, the
bands appearing to be narrower whenever the neighboring
bands are moving rapidly up or down.
The most useful form of band-chart is the “100% band-
chart.” In this case the entire space between the zero or
base-line and the 100% line is filled with various bands, each
CLASS ALIMMEBTS OF THE POPOLATION
Di-irided IntD Capital « Labor ^ and Public
United States
1870-1910
(Source;- Arranged from Census by A. H# Hansen)
Capital
7a
7.7
10*2
io*d
15.8
Public
66*2
63.7
4ea
46.4
41*9
Unolasaified
8a
8.2
9*5
6.6
6*0
Labor
26*6
30.4
32*4
55*3
38.2
1870 1880 18S0 1900
Fig, 194. The Smoothed Relative Band-chart*
1910
COMPOSITE CURVES
215
IBB BAIOBB OF BiCKiKI QOOBS
Pmestio nercbandiso exported classed as consumers* or produoors* goods
United States
1910-1919
(Source;- U, S* Statistical Abstract)
1
Total value g
(|-,000,000)
<
1
s
to
•
o
Ui
1
o
B
cT
o»
-•e
%
o
1M
-c*
inol«iidsoel« g
cT
cT
to
to
Total
.-N
s
to
to
s
— *
i
-«-»*
CD
to
W'
%✓
^ Serai-
5 xafgd
ttBB u>
o
•
«0
.H
a
a
3
«<<4
^ Raw
mat*
Dg
■o
s
to
3
Crude
food
<0
to
a
a
to
I
o
<0
o
0*4 fr»
N r«
a
«o c^ 0»
*«o
u
Fig. 195. The Staircased Relative Band-chart.
Indicating a portion of the total or 100%. The fluctuation
;^pd changes of these hands show graphically the changes of
2i6
CHARTS AND GRAPHS
the component elements of this 100%. This type of chart is
often used to show the changes in the distribution of cost and
profit in an industry. The optional illusion of narrow bands
when nearby bands are moving rapidly up or down is to some
extent eliminated when the band-chart is made with stepping
or staircase outline instead of smoothed polygon outlines. For
IMPOHTS OTO TBS VSITED SIAIBS
ieOO-1920
P®ro 0 ntag« frott the diff«rewi oonttnente
MrioA
Ootasia ®
, Asia ^
:?RsJSSiSaaaajs8a8S2
$ S SI S 9 S S I 8 3 SI I
Fig. 196.
a simple presentation of the changes in the component parts
of any phenomenon, this 100 per cent stepping band-chart is
admirably suited. It is extremely popular in its appeal and
does not suffer from the general disadvantage of staircase-
curves because there is no question of superimposing other
similar charts. It is the right way to represent cost compo-
nents and other percentages to a general public, being within
its narrow limits, safe, sound, and attractive. And the reader
will notice that it is a form of curve which is well-nigh indis-
COMPOSITE CURVES
217
tflUPOKTS Turn THB TJHITBD STATBS
Value
1800 - 1920
percentage to the different continents
Africa •• o o
Ooeaztia ** ^ ® *> ® o«t,^oir^Hrr^NiMeiiii>«»iew
Asia ^
3oath o «« w 40 w <c»04ei<o«^’Cw*e«>'Crti"ir«0'ei«
America
Kortli g ^ ^ «o «e
(America e* h »>< •-<
surope s @ ^ ^
Fig. 197,
tinguishable from a bar-chart, being virtually a vertical-bar
or pipe-organ compound, relative bar-chart, or in other v/ords,
a series of 100% bars set on end and brought into contact with
each other.
In addition to the foregoing more or less distinct types of
curves, there are also many and various possible embellish-
ments which belong to the field of artistic rather than that of
statistical endeavor. The object which is being charted may
be pictured realistically and the picture shown at the end of
the curve. Indeed, the same picture may be used frequently
along the curve, or the picture may be modified to reflect the
changes which are shown mathematically by the curve.
Several different pictures may adorn as many different curves,
and where one rises particularly high, it may be given a pair of
wings or set in a balloon or aeroplane. By these and other
fanciful ways, the imaginative chartmaker may make the
appeal of his chart more vivid. But such measures are out-
2i8
CHARTS AND GRAPHS
MILLIONS
OF
GALLONS
Fig* 198. An Excellent Band Chart (absolute).
Showing the consumption of gasoline by classes of uses. — From Joseph E. Pogucy
Economics of Petroleum,
)910 1911 1912 1913 19H 19t5 1916 191® 1919 1§3<1 192t
Fig. 199# The Relative Chart is Supplementary.
It is well to show data of this kind by two charts, the absolute and the relative
or percentage distribution, and the latter can well be smaller. — From Joseph E,
PoguCy Economics of Petroleum,,
COMPOSITE CURVES
219
191S 1916 1917 1918 1919 ■ 1920 1921
CHANGES IN THE STANDARD OF LIVING.
Index Numbers of Weekly Earnings in New York Factories, of the Cost of Living in
the United States, and of the Living Standard (“Real Earningvs’’).
Fig. 200. A Pictorial Curve.
side the proper scope of this book; the pictorial curve, like the
pictorial bar-chart, is really intended for, and is useful for,
popular consumption. We have come so far into the subject
of mathematical charts that we shall hereafter have no time
for purely pictorial effects. These may be left to the enter-
prise of the individual.
Chapter XX
HISTORICAL CURVES
Statisticians divide all series of figures into two groups.
A series involving time, that is, a series for which different
points or periods of time are the stubs or independent variable,
they call a historical series. Other series in which time is not
the independent variable, they call frequency series. This is
a convenient classification for the chart-maker, and we can
therefore divide all curves into historical curves and frequency
curves. The historical curve has by some writers been called
the ''histogram,'’ or "historigram," and the frequency curve
KEW INCORPORATIONS
Capital Invested in New Enterprises
Whose authorized capital equaled or exceeded $100,000
Principal States, U. S.
1919-1921
(Source:- N. Y* Journal of Commerce)
(Figures in mill ions of dollars)
1919
1920
1921
Jan
492
2280
1243
Feb
324
1169
654
Mar
371
1376
965
Apr
516
1354
988
May
749
1418
601
Jun
1255
1323
676
Jul
1420
1260
282
Aug
823
941
580
Sep
1947
951
490
Oct
2364
1160
503
Nov
1341
896
363
Pec
1078
861
619
Fig. 201, A Historical Series.
220
HISTORICAL CURVES
221
the “pictogram,” but these names have not been widely ac-
cepted in a precise sense.
In historical curves time is always the ^c-variable and must
be plotted on the horizontal axis, its divisions forming the
x-scale. In a previous chapter on plotting points the need
for a precise scale has been discussed and the two methods of
indicating periods of time, either by points on the scale or by
spaces between points, have been dwelt upon. The reader is
urged to review this section, as it meets a serious problem in
the plotting of complicated historical data. The reader is also
referred to a previous chapter on curve scales, in which the
most useful forms and positions of the chart-field were ex-
plained.
The field of a historical curve-chart should be positioned
very close to the righthand edge of the sheet of paper on
Fig. 202. Year by Months, Universal Ruling,
Fig. 203. The Individual Charts Combine Easily.
HISTORICAL CURVES
223
which It IS drawn. There should be not more than a quarter
or half inch margin between the chart-field and the edge of the
paper on this side. The reason for this position will be clear
to you the first time you prepare a series of historical curves
in which the curve travels across sheet after sheet of paper
through a succession of years or periods of time. By over-
lapping the charts (fanning them out) so that they are all vis-
ible, with only these narrow margins between them, the entire
series can be made to appear as a single chart. In this form
the entire series can be conveniently studied and econom-
ically photostatted. The narrow margin between each chart
Serves to break up the curve into its component periods with-
out destroying its continuity. In this way a chart many feet
long can be made upon ordinary sheets of paper without past-
ing them together and without inconvenience in filing or
handling them. And from this long series, a single chart for
a single period can be abstracted and individually compared
with other individual charts. The narrow margin to the right
of the chart may at first seem surprisingly inartistic, but it
pays for itself in the flexibility of uses which it gives to the
chart.
For a similar reason it is well to place the field of a his-
torical curve-chart as low upon the page as possible. You
must of course have room to enter the figures for the horizontal
or A:-scale legibly. This will rarely require more than three-
quarters of an inch. By placing the field low upon the page,
it is possible to compare curves for similar periods of time by
laying one directly above the other, overlapping them ver-
tically so that the two curves are both visible and their ordi-
nates and time scales coincide. By this device, the reader can
easily compare the seasonal or periodic fluctuations of two
curves and at a glance detect the extent of their similarity.
In short, the field for a historical curve should be as close
as possible to the lower righthand corner of the sheet of paper
upon which it is drawn. This leaves a very large margin at
the top of the page above the chart, in which should be en-
tered the data of the curve. It also leaves a large margin to
the left of the chart. This margin will be partly filled by the
important vertical or y-axis scale or scales (if two or more
scales occur on the same chart). But the chief use for the
lefthand margin is that in it can be written the notes, comments
or explanations which may be desired with the chart. If the
HISTORICAL CURHES 225
sheets are to be bound in a loose-leaf holder or book, the
binding edge will be on the extreme lefthand edge still further
away from the chart. At the top of the page above chart and
data, the title should be placed.
In historical curves, possibly more than in most, it is im-
portant that the data appear with the chart. It is important
for the maker of the chart, for the curve is more easily plotted
direct from the data, and the plotting checked for accuracy.
It is important for the reader who is thereby enabled to either
satisfy himself as to accuracy or to find any particular value
without relying upon approximations more laboriously de-
ciphered from the scale. The proper position for the data is,
as has been said, above the chart, each value plotted appearing
on line with the ordinate of its plotted point. Unless the data
is extremely simple and brief and can, without crowding, be
written horizontally, it is better to enter it vertically, writing
or typewriting on edge, in the manner described in a previous
chapter.
In historical curves, we have much use for a few simple
mathematical and accounting phrases. The first of these is
the ^^cumulative,’’ or ‘^total to date.^’ When beside a column
NEW INCORPORATIONS
, Capital Inveatad In New Enterprises
tntioae auttiorlzed capital equaled or exceeded $100,000
Principal States, U. S»
1919-1921
(Source:- N. Y. Journal of Commerce)
(Figures in millions of dollars)
1919
1920
1921
MontMy
Cumula-
tive
Monthly
Coraula-
tive
Monthly
Cumula-
tive
Jan
492
492
2,280
2,280
1,243
1,243
Peb
324
816
1,159
3,439
654
1,897
Mar
371
1,137
1,376
4,815
955
2,852
Apr
616
1,703
1,354
6,169
988
3,840
May
749
2,452
1,413
7,687
601
4,441
Jun
1,255
3,707
1,323
8,910
676
5,117
Jul
1,420
5,127
1,260
10,170
282
5,399
Aug
823
5,950
941
11,111
580
5,979
Sep
1,947
7,897
951
12,062
490
6,469
Oct
2,364
10,861
1,180
13,242
603
6,972
Nov
1,341
11,602
896
14,138
368
7,340
Dee
1,078
12,680
861
14,999
619
7,959
Fig. 205. Simple Series and Annual Cumulations.
CHARTS AND GRAPHS
226
of figures showing the sales of your company, month by month,
you place a second column of figures in which are entered the
NEW mCORPORATIOHS
Capital Invested in New Enterprises
Whose authorized capital equaled or exceeded $100,000
Principal States, U* S#
1017
(Source:- N# Y* Journal of Commerce)
( Figures in millions of dollars)
Cumulative
OJ
to
663
<4*
8
IQ
01
to
1
H
10
0>
to
8
10
s
0
to
10
§
H
01
cT
oT
to
Monthly
0)
H
10
IQ
Hi
01
Oi
10
to
$
lO
01
493
to
V*
1
a
n
03
ca
Fig. 206 .
I
<DOO<1/
& ^ < a ^ ^ 4 SotesQ
1917
Series and Cumulation Plotted With Same Scale.
HISTORICAL CURVES
IV]
total sales to date this year, your second column of figures is
a “cumulative.” The cumulative series begins with zero at
the beginning of each period of time, that is, before the first
value in the period, and is built up by adding each item to the
previous cumulative until the final entry of the cumulative
series is the total for the entire period.* The cumulative for
the next period then begins with a new zero and similarly
builds up to the total for the next period. And it is important
to remember that at the end of each period, the cumulative
has mounted to and equaled the total for the entire period.
We deal here, of course, with periods of time (such as years)
which contain a series of individual values for shorter periods,
such as months.
Not all data can be cumulated. Economists make a dis-
tinction between “stocks” or “funds” and “streams” or
“flows” of goods or money. A stock or fund of goods is some-
thing which can be considered in existence at a certain moment
«e» .•«» «<«»
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•ur 4S.C0
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e«fl SA.tA
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r«> SI.4A
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AB01G.SU.E rttcis 08
^tfisivsR no laoN
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(Aouro* - of Ubor Staitait**)
Fig* 207. This Data Cannot be Cumulated.
^ By the forward cumulative described in the text we obtain “up to and including”
figures. It is of course possible to cumulate historical figures backwards, obtaining
“after and including” figures, but the step seems purposeless. — C/. Secrist, Horace,
Jn Introduction io Statistical Methods, pp. 232, 267.
228
CHARTS AND GRAPHS
or instant of time, while a stream or flow of goods is something
which takes place during a given period of time. Figures of
the latter, that is, stream or flow figures, can be cumulated.
Obviously, if sales have continued throughout the year the
sales for each month can be cumulated, that is, can be added
together to give a total of sales for any period of several
months or for the entire period of the year. On the other
hand, the figures of a stock of fund or goods cannot be cumu-
lated. In business, a common example of a stock or fund is
the stock on hand or balance at any point of time. And
obviously, if your balance was 23,000 on the first of January
and 25,000 on the first of February, you cannot speak of your
balance for the two months together as 28 j 000. It is not
difficult to decide whether a series can be usefully cumulated.
The use of cumulations or series of sub-totals is frequent in
accounting.
The next mathematical conception is at present little used
in ordinary accounting but is far more valuable for most ana-
lytical purposes than the cumulative. It is called the ^^moving
total.’’ To take the example given above, if beside your
figures for the monthly sales of your company, you were to
enter another column of figures showing the sales “for the last
KEW INCORPORATIONS
Capital Invested in New Enterprises
Whose authorized capital equaled or exceeded $100,000
Principal States, U. S.
1919-1921
(Source:- N. Y. Journal of Commerce)
(Figures in millions of dollars)
1919
1 1920
I. 1
Monthly
Monthly
Moving
Annual
Total
Monthly
Moving
Annual
Total
JltTl
492
2,280
14,468
1,243
13,962
Feh
324
1,159
15,303
654
13,457
Mar
371
1,376
16,308
955
13,036
Apr
516
1,354
17,146
988
12,670
Ma7
749
1,418
17,815
60l
11,853
Jun
1,255
1,323
17,883
676
11,206
Jul
1,420
1,260
17,723
282
■ 10,228
Aug
823
941
17,841
580
9,867
Sep
1,947
951
16,845
.490
9,406
Oct
2,364
1,180
15,661
503
8,729
Nov
1,341
896
15*216
368
8,201
Dec
1,07B
861
14,999
619
7,959
Fig. 208 . The Simple Series and Its Moving Annual Total-
HISTORICAL CURVES
twelve months/’ this new column would show the moving
totals. Beside the January sales in 1921, you would enter the
sales of the twelve months beginning v/ith February, 1920, and
ending with January, 1921. Beside the February, 1921, sales
you would enter the total of sales for the twelve months
beginning March 1st, 1920, and ending February 28th, 1921.
It is easily seen that a moving total can be carried all-through
the year by simply taking the total sales for the previous year
and successively subtracting the sales for the thirteenth month
back, and adding the sales for the last month. Each figure in
the moving-total series can be obtained by dropping off one
month in the earlier year, and adding the corresponding month
in the later year.2
The moving total is so useful that it has sometimes been
enthusiastically described as the balance-wheel of commerce.
When you are plotting the curves month by month, you are
apt to find a considerable amount of monthly fluctuations,
due in part only to normal seasonal conditions. In such
cases, these perfectly normal seasonal fluctuations may hide
^ The work-sheets for computing moving annual totals should always be designed
to show similar months (or other parts of the cyclic period) together. This is done
by arranging the periods in successive lines (with like months below each other) or in
columns (with like months beside each other). Space should also be left for two other
figures in each month: the first of these is the moving annual change, that is, the
algebraic difference between the two like periods; the second is the moving total,
that is, the cumulative of the moving annual change.
A second method is to arrange the monthly (or original) data in one long column.
The comparison between like months may then be effected easily by means of a
movable slip of paper with two slots or windows at the appropriate places to make
the two desired months visible and hide all intervening months. The moving change
and the moving total figures can then appear (each in a column) in two columns
beside the column of original data. Totals should be taken directly from the original
data at intervals for checking purposes.
If the cumulative also is being computed, still a different form of work-sheet is
useful. In parallel columns the successive years should be tabulated, with each
monthly figure on every fourth line down the page. The sheet should then be fed
into a listing machine and the tabulated figures reprinted immediately below their
entries. As they are listed, sub-totals should be taken on every record line below
them, these forming the cumulative series. The sheet is then removed from the
machine and the moving annual total entered by hand from a calculating machine
(which adds and subtracts the proper months from the last totals and retains the
results), these being entered in the remaining blank line for each month. This paper
should be originally ruled with horizontal faints at listing machine intervals (of one-
sixth of an inch) and with horizontal heavy lines every fourth line to separate the
months, the page having 48 lines in all. As a further guide, vertical faints can be
ruled in at listing machine intervals (one-sixth of an inch). If not specially printed
up to order, the cheap cross-ruled paper with lines every sixth of an inch can be used.
These details all tend to make checking up for errors very simple, and largely eliminate
mistakes.
230
CHARTS AND GRAPHS
or obscure the true trend of the business, or at least make the
determination of the trend more difficult. But you can easily
tell whether the general trend of your business is upward or
downward by plotting the curve for the moving annual total.
The moving total series appears to flatten out the seasonal
fluctuations and respond only to the true movements of the
trend. In fact the moving total is sometimes called, even by
statisticians, the ^"trend.’’
I The moving total need not be annual, but can be computed
for" any given period of time. Thus we may have a moving
24-months total, or a moving S-year total. In any case we
have again periods within periods, as in the cumulative. The
most usual form is the moving annual or 12-months total,
for ordinarily in business there is a certain amount of normal
monthly or seasonal fluctuation which repeats itself every
year. These annual seasonal fluctuations are naturally swal-
lowed up in a total for twelve months, for such a total always
includes every month in the year.^ The moving total is in
general an excellent device for smoothing out the wrinkles
and wiggles in a curve and reducing the curve to a simple
regular trend-line. It should be used whenever the cycles of
fluctuations appear to be of regular and uniform length or
periodicity.
A word of caution is necessary about the plotting of a
moving total. Strictly speaking, each item in the curve should
be plotted in the centre of the period which it covers. Thus,
the plotting point of each figure in a moving annual total series
would normally be midway between the ordinates of the sixth
and seventh months covered by the figure. The entire period
of the total being one year, each point should be placed in the
middle of the year which it represents. In this case, the moving
® Whenever the period of the annual cycle is not regular, as in crops and tempera-
ture cycles (one period of 123^ or 13 months, the next of IV/i or 11 months) it is
well to follow Professor Secrist’s suggestion of a thirteen-month moving total. This
has the further ad vantage of centering the moving total figure precisely upon a monthly
one (the seventh) instead of midway between two monthly ones (the sixth and seventh).
The same advantages are much better secured by an average of an eleven-month
and a thirteen-month moving total, both centered on the same months. This may
be called a **taper-smoothed"* eleven-thirteen-month moving total, as it gives full
weighting to the central eleven months and half-weighting to the terminal months
(first and thirteenth). It will be seen that this precisely corresponds (in the average)
with the periodicity of eleven to thirteen months. The taper-smoothed eleven-
thirteen-months moving total is easily computed from the twelve-months moving
total, as it is the two-months moving average thereof. Of course, a longer taper can
be used if desired. The test is smoothness of the resulting curve.
HISTORICAL CURVES
231
total curve will begin five and one half months after the
beginning of the curve of individual months, and will end five
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Fig. 209. Three Positions for the Same Moving Total*
and one half months before the end of the monthly curve.
The moving annual total may then be called a ^Total for the
current twelve months,’’ For certain purposes, however, you
may desire to have the moving total end on the same ordinate
as the monthly curve. This can be done by plotting each item
at the end of the year which it represents. It has the advan-
tage of giving a more up-to-date appearance to the chart, but
it is now necessary to label the series moving ^^total for past
twelve months,” On rare occasions, you may desire to place
the moving total at the beginning of its period, in which case
it becomes a ‘^total for the following twelve months.” When
comparing trends, however, between various items, it is im-
portant that these moving totals be plotted at similar points
CHARTS AND GRAPHS
NEW INCORPORATIONS
Capital Invested In New Enterprises
Whose authorized capital equaled or exceeded il00,000
Principal States, IT, S*
1918
(Source:- N, Y* Journal of Commerce)
(Figures In'milllons of dollars)
Moving Total for
Previous Year ^.^xp-«j**oioto*ocjojoj
Moving Total
Current Year rti^t^rorcroTorcJ oToitoto
eouo
Moving Total for S?I
Following Year ci to tfi "fti m
Monthly
Fig* 210. A Detail of the Last Figure.
Pee M U- l - J . I H I I ■■, { ■.■ i - i I . ( . 1 .. L . 1 -L . l l-L - i j 12,680 ' 2,399
HISTORICAL CURVES
233
in their periods, else an unwarranted lag will appear between
the fluctuations of the two charts.^
Similar to the moving total is the ^^moving average/’ The
moving average is merely an average secured by dividing the
1920
1921
Moving
Total
Moving
Average
Moving
Total
Moving
Average
Jan
14,468
1,203
13,962
1,163
Feb
15,303
1,276
13,457
1,120
Mar
16,308
1,361
13,036
1,086
Apr
17,146
1,429
12,670
1,054
May
17,815
1,485
11,853
987
Jun
17,883
1,490
11,206
934
Jul
17,723
1,478
10,228
853
Aug
17,841
1,488
9,867
822
Sep
16,845
1,403
9,406
784
Oct
15,661
1,304
8,729
727
Nov
15 ,216
1,268
8,201
684
Deo
14,999
1,250
7,959
663
UEW INCORPORATIONS
Capital Invested In New Enterprises
Whose authorized capital equaled or exceeded $100,000
Principal States, U. S.
1920-1©21
(Source:- N. Y. Journal Of Commerce)
(Figures in millions of dollars)
Fig. 211. Moving Annual Total and Average Series.
moving total by the number of items which compose it, that
is, a moving monthly average is secured by dividing a moving
annual total by twelve.^ The moving average has the great
advantage of lying at about the same height on the chart as
the curve of individual periods (e.g. months), from which it is
^ Needless to say, the only accurate picture of events (as regards time) displayed
hy the moving-total or average curve is that shown by the curve of the current
twelve months or other period, that is, the curve of points plotted at the centers of
their periods. {Cf. Chapter on Plotting Points, supra) This is the true smoothed
curve. At all other positions, the curve has been arbitrarily “lagged’’ forward or
backward.
^ The work-sheet for the moving average is the same as that for the moving total
already described, save that an additional space must be left in each month (in the
second, or columnar, method, it would be an additional column) for the moving
average, which is derived from the moving total by dividing the latter by twelve
(annually, or by whatever the number of items be which go to make up the total)
(Obviously in the taper-smoothed eleven-thirteen-month moving total, the two
terminal months have only half weight and the total is still of twelve months.)
234
CHARTS AND GRAPHS
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derived.^ And for this reason, you will find it an even better
method of smoothing curves and showing their true trends
than the moving totals
® The moving average has another great advantage over the moving total, in that
it can be given variable period-lengths to conform to cycles of variable lengths.
This is not possible with the moving totals.
^ The moving totals and averages are also sometimes called “progressive” totals
and averages.
1919 1920 1921
Fig. 212. The Moving Annual Average Gives the Trend.
Chapter XXI
CYCLES
Long ago, or was it yesterday, there were neither auto-
mobiles nor aeroplanes, and the streets were frequented by a
cheery and wholesome class of persons, who conveyed them-
selves about on two-wheeled contrivances called bicycles. In
deference to our age, the reader will permit us to pause and sigh
a moment over this happy retrospect. Sometimes in the circus,
a contemporaneous antiquity, trick riders rode one-wheeled
affairs with perilous skill. Needless to say, the rims of these
wheels were as smooth and regular as the circumference of the
averageclock-chart.
Permission of Mr, Walter N, Polakov
Fig. 213. The Mechanical Cyclograph.
236
CHARTS AND GRAPHS
Now a clock-chart— if you have forgotten your early
chapters in this book — is round like the face of a watch.
Radiating lines or radii take the place of ordinates, and con-
centric circles or rings take the place of abscissae. Hence the
chart can be used for the display of recurrent data — that is,
historical data which after a certain period of time repeats
itself. Your only care must be that the period of the cycle
be adjusted evenly and wholly in one complete revolution
about the circle. Then the curve of the data will meet at the
SCASONAL FLUeTUATldN IN O^ERATIOMK
AVERAtt VSAR, X910>X920, IN 25 STaTIS
(SouroAiir- F. F. Dodge Co.)
Fig. 214. As a Chart, This is Worthless.
CYCLES
two ends of the period, forming a continuous and endless
curve. 1
Had the trick cyclist in the circus used a wheel the rim of
which followed the uneven outlines of this curve, he would
indeed have had a bumpy ride. And a line drawn on the wall
behind him, following the shadow of his head, would mark
the same curve plotted on a chart-field of plain co-ordinates.
Study the curve as so plotted in the ordinary way, and you
will see that once every so often the wiggles or^fluctuations
SEASONAL FLUCTUATION IN BUILDING OPERATIONS
Total for 25 States in average year 1910-1920
(Sonroo:- F* W. Dodge Co.)
Millions
of
Dollars:
350.000. 000
300.000. 000
.250,000,000
200,000,000
150.000. 000
100 . 000 . 000
50,000,000
0
Fig. 215. The Rectilinear Co-ordinates Are Much Better.
^ The clock-chart is similar to carp, the fish which is properly prepared by throwing
it away after it has been cooked. When the clock-chart has been well and carefully
drawn, it is ready for the waste-basket. For this reason, no detailed discussion of it
or its polar co-ordinate field is entered into. The only case in which the clock-chart
is a justifiable product is the case of automatic mechanical charts or cyclographs.
These are parts of recording machines for temperature, pressure and the like, in
which a fountain-pen at the end of a pointer leaves an inked record or curve upon the
rotating disc underneath it. They are graphic records, but not otherwise useful charts.
CHARTS AND GRAPHS
238
of the curve repeat themselves, like the digits in a recurrent
decimal fraction. So whenever in a historical curve you detect
the same or similar fluctuations repeating themselves through-
out the curve, you are justified in suspecting that in the
repeated unit or part you have a cycle. It is an important
function in statistical work to detect the presence of these
CIURTS AND GRAPHS
140
cyclic fluctuations, and to be able at will to remove them.
The subject has already been touched upon in the last chapter.
The time between the commencement of one cycle and that
of the next, is called the period of the cycle. This period may
be short or long, according to the nature of the data, a very
frequent short cycle being that of 24 hours or one day. In
business statistics, there are cycles longer than a year. Some
investigators have found evidence of business cycles in which
eras of general prosperity, depression and crises repeated
themselves every four, eight, or even twenty years.^ In meteor-
ological and astronomical sciences, cycles of dry and wet
weather have been found to last thirty-three years and of
warm and cold weather about a hundred years. The most
important cycle in most business statistics is the annual one,
of four seasons or twelve months. The student of business
statistics can almost always assume that he will find more or
less of an annual cycle of seasonal fluctuations. Sales may
repeatedly rise in spring and fall and decline in summer and
winter. Production may then fluctuate somewhat earlier in
the year, anticipating the changing demand. If production
is uniform, ware-housing cycles will appear in order to absorb
the surplusage in low selling periods.
It is not often that the recurrent cycles are identical either
in the shape or the height of their curves. Such variations
may be due of course to incidental and insignificant causes,
but in general studies of broad trade or economic movements,
they are often given a more fundamental importance, as being
significant of real changes in the phenomena studied. The
problem then is to isolate these variations in the cyclic fluctu-
ations, that is, to eliminate from the series its seasonal cycles
and retain its significant changes. To the series which remains
after the removal of cyclic fluctuations, the name of ^‘^secular
fluctuations'’ if often given. And we may therefore look upon
the original series as being a combination of two dilFerent sets
of forces, or movements, which we call, respectively, cyclic
change and secular change. Either or both of these elements
may be the object of your study and it is important that you
should be able to determine them easily.
*The literature on this subject is considerable. In particular, the student should
refer to:
Mitchell, Wesley C,, The Business Cycle, and
Moore, H. L., Economic Cycles; Their Law and Cause.
CYCLES
241
SEASONAL VIRULENCE OF SCARLET FEVER
Uvjmber of Cases reported to Boston Board of Health
1900- 19<H
Fig. 218 . Showing the Use of Relative (Percentage) Figures
and a Rounded Curve.
Indeed, the statistician, and the chart-maker as well, would
fail in one of his most obvious tasks, if he were to report as a
significant rise or fall, a change which was wholly due to its
cycles. Are we to conclude that the telephone business is dis-
appearing because in a series showing hourly number of phone-
calls, our last report is the number of phone-calls between
twelve and one o’clock at midnight? It is true that between
mid-afternoon and mid-night the telephone activity has
dropped off almost entirely, but we must remember that it
does this every night (with the possible exception of election-
day) and that we deal here with a daily cycle. Are we to
conclude that the cold-storage of eggs is a practise of the past,
because our monthly report of warehouse stocks end with
242
CHARTS AND GRAPHS
ACCIDENTS IN MANUFACTOSlNO
Hourly Ooetirrone* of 364 FkIbI end 11,461 Hon-faitl A««id«ntr
llUnola
Throe Years, 1910-1912
(Percentage Plgurea Only)
(Source.- Onlted States Bureau of Labor Statistics)
March, when as a matter of every-day knowledge there is
an annual cycle and the stocks are always low at this time of
the year ?
The subject is more properly a statistical one than a chart-
ing one, but it is of sych importance in the making of the
specialized form of charts which follow that we will outline
briefly some of the simpler methods in use. Our concern here
is with the separation or elimination of the recurrent or cyclic
fluctuations in an historical series. Ordinarily, in business,
the seasonal fluctuation, that is, the annual cycle, is most im-
portant, and the following explanation will be limited to it.
Other cycles may be similarly treated. The most elementary
consideration in the analysis has been made obvious by the
CYCLES
H3
COLD STORAGE HOLDINGS OP EGGS
Stocks of "Case Eg«;s" in Warehouses
United States
1916-1921
(Source;- Survey of Current Business)
(Monthly Average for Five Years, 1916-1920, * 100)
5 yr. Average
1916-1920
192_0_
1921
34 7 0
42 9 1
11 1 1
7 54 151
3 58 139
52 133 166
190
196
185
183
186
173
204
206
195
162
122
71
144
104
49
170
119
66
foregoing illustrations; namely that the relation between the
last item in a series and the corresponding item in the previous
cycle in the same series is of more importance than the rela-
tion between the last item and the immediately preceding
item. In the last illustration, how do this year’s October stocks
compare with stocks of October last year, not how do this
year’s October stocks compare with this year’s September
stocks.
But every business man has progressed beyond this ele-
mentary stage. He asks to see the figures for the previous
month in each of the last two cycles. For he knows that it is
more important to see how the change in stocks from Septem-
244 CHARTS AND GRAPHS
ber to October this year compares with the change between the
same months last year. We may generalize this by saying that
we are now concerned with the relation between the change in
the last two items of the series and the corresponding change in
CYCLES
245
the corresponding two items in the previous cycle in the same
series.® Now it is precisely this relation which the moving
annual total, or average, described in the previous chapter,
tells us. A little study will show that the moving total swal-
K/G PRODUCTION
Recoipto of Eggs at Five Markets
(Bolton, Kew York, Philadelphia, Chicago, and San Franelaco)
United States
1920-1921
(Number of Cases)
' (Source*- Survey of Current Business)
Monthly
Moving
12-Mo.
Average
Monthly
Total
1920 1921
Fig. 222, The Moving Average Shows Trend.
lows up the cyclic variations by the simple process of swal-
lowing up all the various items which make up the cycle. The
^As a matter of fact, the changes of the moving total from month to month are
merely the differences between figures for the new month included and the old month
excluded (one year previous). Hence, the work of computing a series of moving
totals can progress largely at sight by comparing the month to be added with the
month to be subtracted (one year earlier) and algebraicly adding this difference to
the last moving total.
246 CHARTS AND GRAPHS
period covered by the moving total (or average) must of
course be of the same length as the period of the cycle ^ And
the resulting changes in the moving total are merely the dif-
ference between changes in corresponding pairs of months in
the two cycles. For this reason the moving annual total or
average may be called the simplest method of eliminating
seasonal (or cyclic) fluctuations, and determining the true
secular (or long-time) movement.
While for many purposes, the trend as indicated by the
moving total is a sufficient index of the nature of the more
fundamental changes in the phenomenon, yet in the broad
study of economic or trade movements, it still retains too
many insignificant changes. It is true that the moving total
smooths out all the periodically recurrent fluctuations. But
it does not yet yield a simple series of perfectly regular change,
that is, a straight line, or simple mathematical curve, which can
be expressed by a mathematical equation or summarized in so
simple a statement, as that, for example, ‘'the population gains
two per cent annually.” In a much more precise sense the
latter, that is a fitted straight line, parabolic curve, or other
regular series, is called the "secular trend.” It is also some-
times called the "normal” for the particular curve.
Fitting a straight line, or regular curve, to a historical
series, is a matter of mathematical statistics into which v/e
need not go, for it requires skill and judgment to which no
simple rules of procedure apply.^ It is sufficient to say that
when this is taken into consideration the original series of data
which we are analysing can be considered a combination of
three elements, namely seasonal or cyclic fluctuations, a secular
or normal trend, and secular fluctuations. And in the best
statistical work both the former are often removed from the
data before the curves are published. When this is done, the
reader is advised of it by a simple statement to the effect that
the figures published "are corrected for seasonal changes and
normal growth.” Fortunately he has no idea of the problems
involved in this correction.
Accepting then, the moving total or average, as a satisfactory
method of smoothing away all the insignificant and periodically
^ When cycles are of varying lengths, this does not apply, for the moving total
can only be made with uniform lengths or spans. The device next mentioned, how-
ever, the moving average, does not have this limitation and can be made co-extensiv<?
with the cycle.
^ Cf, Chapter on Curve-Fitting,
CYCLES
247
recurrent fluctuations which often make monthly curves un-
satisfactory — a means in short by which we can promptly plot
the trend or general direction of underlying movements in an
historical series — ^we turn to the question of determining the
true nature of the cyclic, that is, the ascertaining of the true
seasonal fluctuations. We wish now not to eliminate the cyclic
changes in the data, but to eliminate everything else in the
data and retain the cycle alone. How can we isolate the cycle ?
The simplest method and one which immediately suggests itself
is to take a single cycle and forget the other cycles in the data.
This gives us beyond peradventure the change within the
lEW IHCORPORATIOKS
Capital Inveated In Kew Sntflpprisft*
f&OSd vitliorised capital equaled or exceeded lIOO.OOO
Principal States^ U. S.
^ lfi21
(Source:- N. Y. Journal of Cormnerce)
(Pigurea In nilllona of dollara)
Jan
Pob
liar*
Apr
Uaj
Jun
Jttl
Au«
Sep
Oet
lOT
Dee
total
1,S45
654
955
988
601
676
282
580
490
505
868
619
7,989
is.es
e.22
12.00
12.40
7.55
8.50
5.54
7.29
6.16
e.si
, 4.62
7.78
100.00
Fig. 223. The Seasonal Cycle Computed from One Cyclic Period.
cycle. If we wish, we can calculate the various months in
this cycle as related to the total for the cycle, that is, change
each month into a percentage of the total for the year, the
latter being 100%.
The trouble with this crude use of a single year as an index
or indicator of cyclic fluctuation is two-fold. For one thing
it does not take any account of secular trend — ^which in the
case of a young and rapidly growing business will be very
marked — and as a result December sales may appear to be
seasonally larger than January sales, though in fact they are
really smaller, because every year the following January sales
exceed the last December sales, just as within the calendar
year the last December sales always exceeded the last January
sales; the result of this error is to skew the seasonal fluctua-
tion curve around in the cycle, tilting up one end of it, giving
us a warped picture of the cyclic fluctuation, in which the
warping or tilting may be so great as actually to shift the
location of the peaks and valleys. If the data be the record
of sales by an individual concrn, no matter how true a picture
it may afford of the experience of the company, it does not
give a true picture of the changes in the market, the seasonal
variations in consumer demand.
CHARTS AND GRAPHS
24S
The simplest way of correcting for the secular trend or
general movement of the phenomenon is to take the months
IhCORPORATIOSS
c»pu*l In Sen Entarprlsss
fhaao authorized capital equaled op exceeded $100,000
Principal States, J S.
1921
(Source - U. If. Journal of Commerce)
(Ptgures in mtlllona of oJllars)
JttIV
Pab
«»r
Apr
U47
Jua
Jul
&ug
Sep
Ool
HOf
Tata
1921
Amount |
1,243
654
355
988
601
676
282
586
490
003
368
619
7,959
Moiring Total i
13,963
13,457
13.036
12,670
U,653
11,206
10,223
9,867
9,406
8,720
8,201
7,959
l?ore«nt«g*
0.92
4.66
'».33
7.01
5.07
6,03
2.75
5.88
S.21
5.76
4.49
7.78
j 71.89
Corraeted |
12,25
10.19
10.07
7.06 1
0.30 ;
3 93
8.19
7.26
6.02
6.25
10.02
lOO.OO
Fig. 224, The Seasonal Computed from the Trend.
not as percentages of the total for the calendar or fiscal year
of fixed span, but to take them as percentages of the moving
total (^"for the current twelve months’^), for the same months.
The results will no longer add up to 100%, but will be less
than 100% if the moving total has fallen and more if it has
risen. The monthh/ percentages of the moving totals must
therefore be summed up and corrected so that their sum
equals 100% (by dividing them by their sum). The result of
this process may be taken as in most cases an entirely satis-
factory record of the typical seasonal fluctuation, during one
year.
r
—
■
■
■
nn
HR
m
"TJO/t
j
TlU^
YEARLY
AVERAGE
. «ro/
(
L
m
jN,
19
17-19.
ASON
ii
AL
N,
m
'-si
m,
r ^
JT""
F
m
■
■
■
■1
— a yo
—1
n
■
9
m
■
■
■
■
iv/9j
-I
■i
M
■
■
1
i
■
■
-20%
■
■
_
□
JAN. F£B. MAR. APR. MAY JUH. JUU AU<3-._ 5EP. „ OCT. NOV.. p£C.
Fig. 225. A Remarkable Case of Changing Cycle Fluctuations.
Typical seasonal changes in interest rates on 60 to 90 day commercial paper for
the years 1890 to 1908 and the years 1917 to 1921. Weekly variations are shown
as percentage deviations from the annual average . — Permission of Mr. Carl
Snyder.
CYCLES
H9
The second objection to the method still remains, however.
This objection is that the cycle is estimated upon a single
year’s experience only. The cycle shown by another year
might be^ somewhat different. And how do we know that
one year is any more representative of actual conditions than
another. Of course, in the case of businesses (or other phe-
nomena) effected by the war (and this includes most business
and economic records), we might quickly throw out the war-
time record, as being wholly unreliable. But in the absence
of special reasons for discarding certain periods or records as
unrepresentative, we may be confronted with many years of
equal significance which yield different cyclic curves. And in
such cases it would be wrong to trust one entirely and discrim-
inate against the rest. The obvious thing to do is to calculate
the seasonals for each of these years, by the method above de-
scribed, and then average them together, to get an average
seasonal. The resulting curve would meet the second objec-
tion and be representative of the entire experience, for which
records are available.
Perhaps the most important use of seasonals in ordinary
business statistics, is the calculation of “quotas,” or planned
“schedules” for the future. In a sales department, for ex-
ample, the quotas assigned in advance to the salesman, or to
the sales districts, should be as fair as possible, and to assure
this, the typical seasonal fluctuations should be known. Our
problem then becomes slightly different. We no longer want
the seasonals most typical of the entire experience of the past,
but we want the seasonals which may be considered most
typical of the immediate future. A simple average of the
seasonals for many years past would give too little importance,
perhaps, to recent developments. It may be that, through
advertising, or through changes in market conditions, the con-
sumer demand has been shifted about in the year (usually to
become more level, that is, regular). For such developments
it is plain that the later years are more truly representative
than the earlier ones.
In the calculation of quotas, therefore, it is well to “weight”
the later years more heavily than the earlier ones before av-
eraging. Ordinarily it is satisfactory to weight each year twice
as heavily as the preceding year. Other weighting systems
can be used, but this particular arrangement leads to a most
easily calculated average seasonal which has been devised and
2^0
CHARTS AND GRAPHS
used by the author for a long time under the convenient, though
somewhat loose, name of the ^^compounded average/’^ It has
the advantage of being easily carried on from year to year
without extensive re-calculations, an important factor in a
busy office, and it also avoids all question of how many back
years to include, by making all except the last four or five
Bhote authorised cepl* el equaled or exceeded $100,000
Principal States, tf. S.
1918-1921
(Source - U. Y. Journal of Coramorce)
(PlBores In nllllcins of dollars)
Jan
1 Pet
Mar
Apr
May
Jun
Jul j
Aug
Sep
Oct
»ov
Dee
lota!
asis
Percentage !
6.28
4.12
4.69
6.35
8.05
6.23
6.67
4.77
7.24
4.91
6.27
5.42
68.90
Corrected
9.12
6,98
6.80
9.21
11.70
9.04
8.08 j
6.92
10.51
7.13
7.64
7.87
100.00
1919
Percentage
18.87
11.79
12.71
16.22
20.68
27.06
24.16
12.63
23.49
22.44
11,43
8.62
209.88
Corrected
9.00
6.62
6,08
7.74
9.86
12.80 j
11.50
6.98
11.20
10.71
5.46
4.06
100.00
total
1910 and 1919
18.12
11.60
; 12.88
16.95
21.66
21.84 '
19.58
12.90
21.71
17.84
13.09
11.93
200.00
Average
9.06
6.80
1 6.44
8.46
10.78
10,92 1
9.79
6.45
10.81
8.92
6.54 1
6.97
100.00
1920
Percentage
15.76
7.57
8.44
7.90
7.97
7.41
7.11
6.27
8,64
7.54
6.68
6.76
92.24
Corrected j
17.10
8.21
9.16
a.57
8.62
8.04
7.71
5.71
6.11
8.17
6.37
6.23
100.00
•iotal
1920 and Average I
26.16
14.01
15.60
17.03
19.40
18.96
17.60
12.16
16.92
17.11
12.91
12.20
200.00
Compound Average j
13.08
7.00
7.80
8.62
9.70
9.48
8.75
6.00
8.46
8.86
6.46
6.10
100,00
1921
Percentage {
8.92
4.86
7.33
7.81
6.07
6.03
2.75
5.8P
5.21
5.76
4.4°
7,7H
71.89
Corrected
12.26 j
6.77
10.19
10.B7
7.06
8.39
3.93
8.19
7.26
8.02
b.26
10.82
100.00
Total
1921 and Average
25. S3
13.77
17,99
19.39
16.76
17.87
12.68
14,27
15.72
16.67
12.71
16.°?
200.00
Compound Average
12,66
6.89 j
9,00
9.69
8.38
8.93
6.34
7,14
7.86
! 8.29
b.36
a.4b
100.00
Fig. 226 . The “Compounded Average” Seasonal.
negligible. The trick is to average the seasonals for the first
two years (which can be done at sight), and than average the
resulting average with the next year to get a new average,
continuing this process through the years and always working
by inspection.
With the method here outlined to use when you wish to
ascertain the seasonal fluctuations in your data, using the
®Of course, in this ^‘compounded average"’ the weighting is not two to one for the
first two years, but with the exception of the first year the weighting is in this ratio
throughout, and in a very few years the importance of the first year is rendered so
negligible as to be lost.
By other weighting systems, it is meant that ratios of three to one, or of one to
two-thirds, or of one to three-fourths, or the like, can also be easily used and currently
maintained (that is, brought up to date) almost by inspection.
The theory of the compounded average is very simple, and appears to be sounder
than that of any fixed average seasonal. It is believed to be an original contribution
to the science of averages, which should have particular value in economic work with
phenomena undergoing changes in seasonal fluctuations. A very spectacular case of
such a phenomenon was the behavior of the interest-rates for loans in New York
after the establishment of the Federal reserve system, when a previously marked
seasonal was almost entirely wiped out in a few years. The compounded average
affords a sort of moving or progressive seasonal well adapted to such cases. And, in
the ease with which it is brought up to date, it is, mechanistically, a decided labor-
saver and time-saver.
CYCLES
251
“compounded average” in the place of the simple average for
quota-making, and with the moving totaF and average previ-
ously described for the elimination of the seasonal when you
wish the real underlying movement, loosely called the secular
trend, in your data, you are equipped with the mathematical
means necessary for the successful use of the following charts.
Apart from the need of the cyclic change in quota-making,
the usual need is for the secular trend and the latter is indeed
useful not merely for the following charts, but for a wide vari-
ety of purposes in statistical work. It is therefore the more
important trick to have up your sleeve, in attacking either
business or sociological statistics. However far it may fall
short of a true secular trend, it still gives a significant and
easily understood smoothed curve. Though still little known
to the average executive, it is proving extremely popular among
those who use it, and has been credited by some business sta-
tisticians, chiefly those who use the device described in the
next chapter, as being the only part of the data in which the
executive should be interested.
^ By an oversight, the tables in the discussion of the compounded average all
show the months as percentages of the moving total for ‘"previous’’ 12 months. It
is obvious that the moving total of “current” 12 months should have been used.
Chapter XXII
ZEE-CHARTS
In this chapter we enter the accountant’s paradise, and in-
stead of simplifying our data and presentation, we multiply it
three-fold, by adding to each original series of data, its cumu-
lative and its moving total series. The result is a chart which
shows simply and coherently everything about the data which
can be shown. It takes much space, for each important
period (i.e. year or month) of data should be given a separate
chart, and its use is therefore better restricted to a few series,
whose importance is sufficient to justify their treatment in
this thorough and painstaking way. In its way, this chart is
the last word in the analysis of and research into past his-
torical data.^
The ^^Zee-chart” gained its name from the fact that its
three curves roughly form the letter ‘‘Z.” These three curves
are, first, the curve of the original data, second, the cumula-
tive curve, and third, the moving-total curve. In common
practice there are three kinds of these charts, depending upon
the time period of the original data. Where the original data
consists of monthly figures, the chart shows twelve of these
figures to form one year; when the data is weekly, fifty-two
weekly figures are combined in one chart to show one year;
and when the data is daily, thirty or thirty-one days are com-
bined to show in one chart a month. In the first two cases
the moving-total is an annual one and in the last case it is a
^ The Zee-chart is rumored to be of German origin, but appears to have had a
somewhat later independent American discovery. It has received its greatest develop-
ment at the hands of Mr. Willard C. Brinton, consulting engineer and author of
Graphic Methods for Presenting Facts. The present writer is informed that the combi-
nation of monthly and moving total curves on standard scale combinations was worked
out by Mr. T. R. Robinson, and the addition of the cumulative curve was suggested
by Mr. Wallace Clark. Of late, the Zee-chart has been further modified in its form
by Mr. Arthur R. Burnet who has also suggested the omission of scale-figures to
focus attention upon the curves, and has invented scale-finding machinery to facili-
tate plotting,
25a
ZEE-CHARTS
253
monthly one, though it is to be noted that in most businesses
a monthly moving-total has little significance. The Zee-charts
can, however, be made up of any combination of time units
desired. The most practicable one, and the one which will be
herein described, is the Zee-chart of monthly data, twelve
months comprising one chart. Its ready adaptability to the
needs of the business man and accountant makes it extremely
useful for recording data in these fields.
Fig. 227. A Year by Weeks.
Showing the arrangement and methods advocated by Mr. Burnet.
By this time, if you have understood the last two chapters,
you will be protesting that it is not practicable to show a
moving total curve on the same chart with the curve of the
original series. Why? Because the moving total for twelve
months is twelve times as great as the average monthly items,
and if the moving total curve is to be shown, the curve of
monthly figures will lie very close to the bottom of the chart.
Fig. 228. Four Zee-Charts Forming a Single Series.
Notice the diiFerent scales for monthly and annual data and the corresponding positions of the data captions. In the originals the annual moving
total and cumulative data and curves and scale are in red.
ZEE^CHJRTS
^55
its fluctuations hardly visible. The objection is well taken,
but the Zee-chart cleverly dodges this difficulty. How? By
the simple device of using two scales. A large scale is used
for the original monthly figures, but a small scale is used for
the cumulative and moving total curves, whereby they are
brought down on the chart to not more than two or three
times the height of the monthly curve. The ratio between
these two scales has been more or less standardized, the
monthly-curve scale being five times as great as the annual
cumulative and moving total scale. When the data are weekly
and the moving total is a S2-weeks total, the ratio is larger,
the weekly scale being twenty times as great as the cumulative
and moving total scale. When the data is daily and the moving
total monthly, the ratio is again diflFerent, the daily scale being
ten times as great as the monthly cumulative and moving
total scale.^
Standardized practice also has it that a color distinction
should be observed between the two scales and their corre-
sponding curves and data. You should enter the individual
series in the data at the top of the chart in black, plot the
curve therefore in black and enter the scale itself in black
lettering. But the cumulative and moving-total figures should
be entered in red in the data at the top of the chart, likewise
their curves should be drawn in red ink and their common
scale entered in red letters. This distinction is an excellent^
one as it causes the curve of original data to stand out most
prominently while the moving total curve or trend is perfectly
clear, some distance higher on the chart.
As you have seen in an earlier chapter, the moving total
of a series can be plotted at any point within its period, and
can be a moving total either of the current twelve months, of
the following twelve months, or of the past twelve months.
It is the last kind of moving total alone which is used in the
Zee-charts. From this fact, two advantages arise. In the
first place all three curves are entered at the same time on the
same ordinate and a hasty reader who has no time to analyze
the figures, gains no false impression that the chart is not
A very shrewd suggestion has been made by Mr. Arthur R. Burnet, consulting
statistician and graphic expert, that the scale-figures be omitted from charts which
are to be shown to the non-technical business man, in order that his attention may
not be diverted from the behavior of the curves. The fact that data is always in-
cluded in the Zee-chart, and hence scales can always be ascertained though not cali-
brated, gives to this suggestion unusual merit.
ZEE^CHARTS
^S1
thoroughly up-to-date, as he is likely to when the moving
total is plotted upon an earlier ordinate. In the second place,
the cumulative and moving total curves always come together
at the end of each chart by this device, since it is obvious that
the cumulative for twelve months is the same as the total for
the past twelve months. This coincidence of the two curves
is an important element in the simplicity of the chart and also
automatically checks both computing and plotting.
The Zee-chart should be prepared upon the same type of
paper as other historical curves, that is, its field should be
positioned in the lower righthand corner of the page, close
to the edge of the paper. For Zee-charts are generally pre-
pared in sets for data extending over many years and requiring
as many charts in the set as there are years. It is important,
therefore, that the reader be able to fan the charts out, each
overlapping the next so as to afford a general view of the entire
series of years in a small space. The narrow margin to the
right of the chart on each sheet produces gaps between the
Permission of Mr. Arthur R. Burnet and the Ronald Press.
Fig. 230. Two Charts Fanned Upward to Study Seasonals.
CHARTS AND GRAPHS
258
charts when laid out in this way but these gaps are of benefit,
forming slight breaks between years without destroying the
general continuity of the set of charts. The low position of
the field upon the paper facilitates the study of seasonal fluc-
tuations by overlapping charts one above the other. Needless
to say, the scale on all charts belonging to a set, should be
uniform and rigidly maintained within the set in order that
the curve running through the charts may be homogeneous.
The scale on a Zee-chart should be somewhat smaller than
on most historical curves, for the reason that you probably
expect to continue the Zee-chart series in the future, and you
must allow ample margin for future growth of the business
and future peaks which will rise higher than the past peaks.
To find the scale for a Zee-chart, therefore, look back over the
series of moving-total figures in your data and locate the
highest peak figure in the past, and then select a scale which
Fig. 231. Table for Zee-chart Scales,
For standard chart-fields six inches high.
will bring this figure about half way up the chart. Selection
of the scale is perhaps one of the most difficult parts of Zee-
chart making, and it is convenient to use a table of scales
similar to the table for historical scales given in a previous
chapter, modified onl}?* by increasing the key numbers one-
third to lower the peaks to half way up the chart.
One of the most interesting features of Zee-charts is their
use in what is called ^Tght analysis.’’ Light analysis is a
method of comparing two curves or charts drawn on similar
scales. You place the two charts together, one upon the
ZEE^CHARTS
259
other, and hold both up to the light- Both curves will then be
visible, enabling you to compare them minutely. In some offices
where a large number of Zee-charts are used, a machine called
a ‘‘light box’’ is kept for this work. The light box consists of
an electric light underneath a piece of ground glass, which
throws a uniform illumination up to the paper laid upon it-
The charts should therefore be made on highly translucent^
paper, and the field should be positioned with absolute uni-
formity on every sheet. These are considerations which hold
not merely for Zee-charts but for all curve-charts to be sub-
jected to light analysis.
A number of attempts have been made to adopt the Zee-
chart to current operation control. It is easily seen that by
extending or projecting the cumulative curve on any up-to-
date chart in which the end of its period (e.g. year) has not
yet been reached, an estimate can be quickly made of the
probable amount to which the future months will bring the
entire year’s total. Moreover, if a quota for the year has been
made in advance, the cumulative for this quota can be easily
plotted in pencil or, best of all, in yellow ink and a comparison
between the red cumulative for actual performance with the
yellow quota cumulative will show how well or poorly the
quota is being fulfilled. A slightly different method is that of
showing the quota cumulative as a straight sloping line, and
entering the curve figures and cumulative as percentages of
the amount which would have been necessary to meet this
quota. But these various methods have not been so successful
as to find general acceptance, as they tend to fill up the chart
with too much detail.
The plain fact is that the Zee-chart is a “looking-backward”
chart. The best that can be said of it is that it is an ideal
method of historical research. The sales of your chief line or
department should be plotted in this way, in order that you
may study their past history to the best advantage. The chart
is designed to show you at once the individual monthly or
periodic fluctuation, their general trend or moving total, and
the cumulative or total to date for each year, and from these
three accounting elements for each year, you can see just when
sales began to fall off or rise, what the seasonal fluctuation
was, and how each year’s individual progress compared with
The paper may be highly translucent without being in the least transparent.
26 o
CHARTS AND GRAPHS
Fig. 232. Mr. Burnetts Arrangement.
that of other years. The Zee-chart is, among amount-of-
change charts, the last word in historical research and detail.^
^ An excellent description of the Zee-chart is to be found in a series of articles by-
Mr. Arthur R. Burnet in Management Engineerings beginning Sept. 1921, pp. 153-160.
The first American description is that by Mr. John Wenzel in an early issue of
the Scientific American Magazine,
The best adaptation for forecasting and quota-measuring has been made by
Mr. John Scoville, Maxwell Motors, Detroit.
Chapter XXIII
PROGRESS CHARTS
From the paradise of the accountant and historian let
us step into the paradise of the operating executive. The
operating executive is the man who sees to it that things are
done. His interest begins and ends with the job to be done,
how much of it has actually been done, and how much remains
to be done. When tPld that a job has not yet been finished,
he is not interested in excuses and reasons, he is not interested
in why or when the performance fell below the schedule, he is
only interested in the amount remaining to be done and the
job of getting it done. We shall not expect to find him satisfied
with a retrospective or “looking-backward” chart. For him,
the “looking-forward” chart!
F ar and away the most “forward-looking” chart known is
the “progress” chart. It is the product of the man who was
probably the greatest engineer America has ever produced,
the late Mr. H. L. Gantt.i It is significant that this chart
method was devised by an engineering type of mind, for it is
admirably adapted to an executive control of operation.
Compared with its dynamic influence on the actual control of
operations, nearly all other types of charts seem to justify
the assumption that the word “statistical” is derived from the
word “static.”
^ Henry Laurence Gantt was born in Maryland on May 20, 186L He received
the degree of bachelor of arts in 1880 from Johns Hopkins University and in 1884
the degree of mechanical engineer from Stevens Institute of Technology. He died
November 23, 1919, at his home in Montclair, N. J.
Mr. Gantt was associated with Frederick W. Taylor in his early work at the Midvale
and Bethlehem Steel Companies and a few years later established his own consulting
practice as an industrial engineer, which he carried on until his death.
Among his clients were many of the most advanced manufacturing companies of
this country. During the war he devoted his entire time as well as that of his stalF
to the solution of the Government’s problems of production and management. He
acted in a consulting capacity for the Ordnance Department, the Naval Aircraft,
the Shipping Board, and the Emergency Fleet Corporation.
Mr. Gantt developed a method of paying workmen according to the results they
261
263 .
CHARTS AND GRAPHS
The Gantt progress-chart presupposes a definite detailed
schedule or plan made out in advance and generally called the
‘^quota/’ The chart itself merely measures the subsequent
actual performance, when it takes place, against this pre-
determined schedule or quota, and shows emphatically
whether or not this quota is being met. The chart shows
incidentally how much of each past month’s quota has been
accomplished, but primarily it shows how much of the cumu-
lative or total quota to date has been accomplished. The chart
never shows trend or moving total, nor does it necessarily
show even the individual monthly figures; its main function is
to show how much of the schedule has been performed up-to-
date and how much remains to be done.
As we might have expected in a chart for an operating
executive, the progress-chart compresses its information into
very small space. Where the Zee-chart expanded every series
of figures three-fold and used a separate sheet of paper for
each series, the progress-chart combines twenty, thirty, or
even more, series upon a single page. An entire business or
industry can be summarized upon one of these charts, each
of its thirty or more items being in turn shown in detail with
thirty or more sub-divisions on subordinate charts. In the
course of time, the Gantt progress-chart will come to be
recognized as the sine qua non of management, whether it be
sales management, office management, or production and
factory management.
Strictly speaking, the Gantt progress-chart is not a curve-
chart at all. It is rather a horizontal bar-chart, very peculiarly
constructed. But the relation between bar-charts and amount-
of-change curve-charts is so intimate that it can best be
examined here. If you like to so consider it, the Gantt
progress-chart is a combination of many curve-charts, each
flattened out into one dimension and all placed close together
accomplished, which is known as the Gantt Task and Bonus; he developed the theory
that the cost of an article includes only those expenses actually incurred in the pro-
duction of that article, and that the expense of maintaining one machine in idleness
can not be charged into the cost of the output of another machine. In accordance
with this theory, he worked out a method of arriving at costs of idleness and of work;
he originated the Gantt Chart, which compares the amount of work done in a given
time with what should be done and emphasizes the reasons for failure to attain that
stand aid; he introduced a change in the installation of management methods from the
old type, which organized from the top down, to a new type which builds from the
bottom up. Work, Wages and Profits'' (1910), “Industrial Leadership” (1916), and
“Orj[>anizing for Work” (1919), are the titles of the most important books written
by Mr. Gantt . — JVuUace Clark.
PROGRESS CHARTS
26 j
on a single chart. Or if you prefer, the Gantt progress-chart
is a series of horizontal bar-charts with cumulative bar-charts
superimposed.
The scale of the progress-chart is one of its most interesting
features. At first glance, there appear to be as many scales
on the progress-chart as there are bars, or items. That is to
say, every bar on the progress-chart appears to have its own
scale or set of values for the horizontal distances through
which it passes. And, unlike all other charts, the horizontal
distances appear to have no equal and uniform values through-
out the length even of a single scale; on the contrary, the values
given to equal horizontal distances, or spaces between vertical
lines, seem to change weirdly from space to space throughout
each individual line. But the secret of the puzzle is very
simple. The common and proper scale for progress-charts is
“time.” Each equal distance represents an equal unit of time,
and you will find a time scale placed at the top of each progress-
chart, one single uniform scale for the entire chart. Time, then,
is the measure or unit of measurement against which perform-
ance is measured. In Mr. Gantt’s words, “time is the one
common thread running through all operations,” and time is
therefore the one basis on which all performance should be
judged by executives.
Now it is obvious that the number of dollars worth of
goods sold during a month cannot be taken directly as a part
of, or measured directly in terms of, the number of days in
the month. The amount of sales, production, or other per-
formance, and the length of time involved in the performance
are numerically incommensurable quantities. We must there-
fore have a ratio or co-efficient between the two, that is, we
must make up our minds that a certain unit of time is to equal
a certain volume of sales or other performance. Then the
actual amount of performance can be judged in terms of this
predetermined quantity, which has been decided upon for the
given period of time. And these predetermined quantities are
the items which at first glance appeared to form the irregular
scales for each bar.
In short, the progress-chart measures actual performance
in terms of a standard. This standard is shown by the small
figures in each space and is graphically represented by the
entire space itself. Actual accomplishment is graphically,
recorded by a bar drawn across the part of the space which
PROGRESS CHARTS
265
corresponds to the percentage (of the standard), which has
been accomplished. In other words, the standard for each
space (or period of time) is considered to be 100% for that
period of time, and the actual accomplishment during that
period of time is taken as percentage of this standard, and
shown by a 100% bar, in which the shaded portion represents
the part accomplished, the unshaded portion the part not
accomplished. This is literally true of the light lines or bars
which begin afresh at the beginning of each new space and
indicate the monthly or periodic performance or accomplish-
ment. The method is not quite so simple, however, in the
case of the cumulative performance or accomplishment, that
is, the accomplishment from the beginning of the entire
period shown on the chart, to date. This cumulative of per-
formance is shown by heavy bars. Here (1) all the standard
cumulatives which are less than the accomplishment cumu-
latives are considered wholly performed and 100% done,
and the heavy cumulative bar is drawn entirely across them;
(2) the remainder of the accomplishment cumulative, after
subtracting the last standard cumulative, is taken as a per-
centage of the next individual period standard and (3) the
cumulative bar is drawn correspondingly across the corre-
sponding percentage of the last period space.
A simple example will make this clear. Suppose we are
allowed by the publishers of this book ten months in which
PROPOSED
SCHEDULE
Month Quota
1 3
2 1
3 A
4 6
5 6
6 6
7 e
8 6
9 6
10 -A.
Total 50
Fig. 234.
to prepare F . As a matter of fact, the book is the result of
many years of study — but that is another story. Being a
PROGRESS CHARTS
267
methodical sort of person, we sit down and prepare an outline
of the book, and discover that it will take about fifty chapters
in which to tell all that you should know about the subject.
We then prepare a schedule showing how long it will take us
to write each chapter, and decide that we can write the first
three chapetrs in the first month, one chapter in the second,
then four and thereafter six chapters a month. Next we pre-
pare a progress-chart of this work showing ten months on the
chart, one space for each month. We enter the number of
chapters to be done each month in the upper left hand corner
of the space for each month. We also enter the cumulative in
the upper right hand corners of each space, showing that at
the end of the second month we will have written four chapters,
by the third month eight chapters, by the fourth, fourteen,
and so on, until at the end of the tenth month fifty chapters
are written. This checks our monthly schedule.
Time passes and we are writing, patient reader — though you
might not guess it — we are writing with meticulous and pains-
taking care. If at the end of the first month, only one and one-
Fig. 236 . The Chart on Jan. 31 st.
The light line records a performance of 50% of the month^s quota; the heavy line
records 50% of the first month’s quota cumulative. The V-shaped mark shows
date of last entry.
half chapters have been completed halfway across the first
space, when we should have written three chapters, we draw
two lines, one light and one heavy, the light one being above
the heavy one. By this we know that at the end of the first
month we have only done 50% of the month’s quota. In the
second month we finish the second and also a third chapter.
A new, light or monthly bar can be drawn all the way across
the second space and a second light bar halfway across, above
it, showing that we had done our bit and 50% more in the
second month. But as we were short in the first month’s
work we are still short to date, having done three chapters
268
CHARTS AND GRAPHS
when we should have done four. These three chapters finish
our quota for the first month only and leave nothing accom-
plished out of the second month’s quota. We therefore draw
the heavy cumulative bar completely across the first space
Progreaa Chart
Jan
P»b
tear
Apr
Jim
Dietatloo
5
-
3
1
4
4
*8
6
T
"e
*6
TP
■
■
■
"
I
1
i
Fig. 237. The Chart on Feb. 28th.
The light lines show 150% of the second month’s quota performed in the second
month; the heavy bar shows total performance to date (Feb. 28) just one month
behind schedule.
but not into the second space at all. A glance at the chart
now shows us that we are short one month’s work (though we
have exceeded our quota in the second month). If in the
third month, in an excess of energy, we write six more chapters,
we shall have done 150% of the third month’s quota of four
chapters and shall therefore draw a light line all the way across
the space, and over it another light line one half of the way
across the space. But the cumulative or total to date will be
nine, enabling us to draw the heavy bar all the way across the
page to the end of the third month and one-sixth of the way
across the fourth space, since the fourth months’ quota was six
chapters, of which we have done one. A glance at the chart
now shows us that we are ahead of schedule. This illustration
may seem wholly personal, but the sales or factory manager
who cannot see in it a method applicable to his own problems
is not worth his salt.
You will now understand something of the unique merit
of the progress-chart. It has an uncanny power of making
human judgment. It does not merely record what has been
done, but in addition thereto it records whether this accom-
plishment has been good, bad, or indifferent. It does not
merely put the question, but it also gives the answer. It
weighs every fact in the balance and states in unmistakable
terms the judgment. If you have fallen down on your job,
the chart does not waste emphasis on why, when, or how you
fell down, but places before you in a way which you cannot
dodge the fact that you have fallen down. It is of course
PROGRESS CHARTS
true that the chart shows when you fell down, and you can,
by notes, comments, or various symbols enter your excuses
upon the chart, but .first and last you cannot escape the fact
that you fell down on the job. Likewise, if you have done
better than vour task (standard, quota, expectation, or what
SALES -ARTICLE A.-Car-+ons
CHARTS AND GRAPHS
Fro77i 7F allace ClarFs^ ^^The Gantt Chart*\ published by the Ronald Press ^
Fig. 239. A Progress-Chart of Sales by Districts.
PROGRESS CHARTS
271
you will) the chart will bear emphatic witness to that fact; it
will proclaim your success from the housetops.^
Picture to yourself a busy sales or factory manager receiv-
ing the usual detailed report on the production or activities of
his various departments. The record is in tabular form show-
ing what each department has performed during the month,
or during the year to date. Before he can decide whether
the work is satisfactory, he must study each figure, and in
the light of his knowledge of all the various factors and cir-
cumstances, decide whether each item shown is satisfactory
or not. It is a task which will cost him many hours of close
concentration on every occasion when the report is submitted
to him, and each time he will find difficulty in remembering
just what were his previous decisions about each item. An
executive’s time is being taken up, not in getting things done,
but in thinking about them. It would not be so bad if it
could be accomplished once for all; the pity of it is that it has
to be repeated every time a report is submitted to him.
Now let us help this executive by giving him a Gantt
progress-chart at the beginning of the year or period for which
the chart is to run. Let us sit down with him and ask him,
once for all, to consider the various factors which in his judg-
ment will be the basis of satisfactory work during the year to
come, and in the light of those factors to determine upon a
reasonable standard for the coming year’s activity. Often he
will give us merely the salient factors and their approximate
influence and leave to us the working out of a detailed schedule
in accordance with them. Often we will get much of the
detail from subordinates closely in touch with them. In any
case, what we are going to try to do is to devise for the entire
coming period a schedule of reasonable expectation of the
business for the coming period (e.g. year) worked out in detail
for each element, department, or other subdivision, and for each
unit of time (month, week, or day). This reasonable standard
or schedule of expectation is sometimes called a “quota” or
“task.” It will often work very well as a quota, or basis of
rating by merits or demerits, subject, of course, to unforeseen
® “Unlike statistical diagrams, curve records, and similar static forms of presenting
facts of the past (Gantt) charts ... are kinetic, moving, and project through
time the integral elements of service rendered in the past toward the goal in the
future.'* — ^Walter N. Polakov, Principles of Industrial Philosophy, Proceedings of the
American Society of Mechanical Engineers, December, 1920,
CHARTS AND GRAPHS
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changes in the various factors or circumstances. But the name
is not important, and whether we call this a quota, a standard,
or a schedule, the point is that we have reached what the
PROGRESS CHARTS
executive considers to be the best basis for the judgment of
work done. And we have finished the major part of the work
of preparing a Gantt progress-chart. We enter these figures in
the blank form and wait for performance to show the accom-
plishments as they occur, graphically on the chart.
Sometimes the difficulty of preparing a reasonable quota is
so great, or the factors and circumstances which will be in-
volved in the work are still so obscure, that a quota is not
desirable. Nevertheless, you will find the executive later on
using some figures or other for comparison. Most frequently
he will be using the figures for the previous year as a basis of
comparison. So when we cannot make up an ideal standard,
we merely use the last year’s record, or perhaps an average
of several recent years, in the place of a quota on the chart.
Gantt engineers, working on production and office problems,
have developed a scientific technique in quota or schedule-
making which is intimately connected with and greatly simpli-
fies the problem of cost accounting in the plant. But whether
these scientifically reliable quotas, or merely rough guesses, or
simply the previous records, be used as the standard, it is all
one to the progress-chart. The chart will use any basis
adopted, and judge the performance in terms thereof.
Note the time-and-labor-saving value of the progress-
chart. After the fundamental decision as to schedule standards
have been made, the chart works automatically. It is a ma-
chine, passing up to the executive his own judgments. All the
labor involved has been transferred to clerks. The executive
merely glances down the chart, noting the length of the heavy
bars. His attention is immediately drawn to the exceptional
performances which he would wish to study. There is no
dodging or forgetting these exceptional cases as shown on the
chart, and the executive is enabled either to discount the ex-
ceptional cases in the light of further developments or unfore-
seen circumstances, or to take immediate action where such
action is called for.
This method of charting is so simple that Gantt engineers
are accustomed to install it in shops for the use of foremen,
as well as for the central planning and executive departments.
They are accustomed to enter the quota in ink and the graphic
bars in pencil (black lead pencil). The maintenance of these
charts requires no special staff of draftsmen or computers;
they are used and filled in by the workmen and foremen them-
LOAD CHART FOR HAMMERS
PROGRESS CHARTS
selves, the bars being roughly sketched in without regard to
nice appearance. Where the charts are used in a statistical
department, or for the higher officials of a concern, it is perhaps
better to observe more care in the appearance of the chart,
and in this case colors can be used to advantage. The monthly
or individual time unit quotas can be typed on the chart in
black typewriting and the cumulatives in red; the light bars can
be ruled in black India ink with a drawing pen, the heavy
cumulative, in red waterproof ink. If this color distinction is
observed, the ends of the red bars each month should be marked
with light black cross-lines vertically cutting the red bar into
segments, and showing the position of the cumulative at vari-
ous times in the past. The initial letter in the name of the
month can also be entered in black on the red bar to identify
these points. When no color distinction is made and the
cumulative bar is black, it is better to draw slight notches
below each final position of the cumulative bar to keep this
record of cumulatives. The first line upon a progress-chart is
usually used for the total of the other lines and its bars are
drawn with extra wide or heavy rulings, for the sake of em-
phasis.
A further refinement has been adopted by Gantt engineers
in the binding of these charts. The charts are bound at the
righthand edge of the paper in order that the stubs at the left-
hand side may be quickly seen. This is of benefit where a large
number of progress-charts are kept in bound volumes.
Ffom Wallace Clark's ^^The Gantt Chart'\ published by the Ronald Press.
Fig. 242, on a Shprt Fly-sheet — the Gantt Way.
276
CHARTS AND GRAPHS
It has been said that the data should always be available
with every chart and the Gantt progress-chart is no exception.
Performance or accomplishment is portrayed graphically by
the bars, but the figures which these bars represent are not
shown on the chart. A blank form, similar in its ruling to the
form of the chart, is laid immediately above the chart page
with the column for stubs removed, so that the page will be
short and the stubs as entered on the chart will be visible for
both chart and data sheet. On this blank form the data of
performance is recorded in writing, the figures for both indi-
vidual time units and cumulatives being shown in the same
spaces as used for their corresponding quotas on the chart.
Because this method presents the data rather than the chart
itself when the page is first opened, and the data sheet has to
be turned over to read the chart-sheet, the winter’s individual
practice is to reverse the arrangement and place the chart on
the short sheet and the data on the full sheet underneath —
an act of heresy which Gantt engineers are not expected to
endorse.
It is with reluctance that we leave the subject of progress
charts. So far as the executive interested in the graphic con-
trol of operations is concerned, the entire subject of charts and
graphs begins and ends with this chapter. No other method
of charting has yet been invented which presents the essen-
tials of operating so forcibly, so clearly or in such small space.
The details for each department- can be shown on depart-
mental sheets with the total for the department at the top
of each chart. The various department totals can be brought
together upon a single plant sheet which will show to the
plant manager at a glance his various departments, and the
total for the plant, carried at the top of his chart. Likewise,
on a single summary chart for the entire business, the presi-
dent or controller can see the work of the various plants, with
a total for the entire business at the top of his chart. In
other words, this method of charting can be carried to the
last degree of detail or reduced to the shortest possible sum-
mary, and the entire structure of an industry can be shown
by a similar structure of co-ordinated charts. Once started,
the work can be carried out almost entirely by clerks and
secretaries, freeing the executives from routine, analytical
work, and largely freeing any special statistical department
for research work. The progress-chart is the ^looking-forward’"
PROGRESS CHARTS
chart par excellence, beside which all other charts are either re-
search methods or ^looking-backward’^ records.^
® The best descriptions of the Gantt charts are to be found in Mr. Wallace Clark’s
The Gantt Chart, Ronald Press, New York City.
The following articles in periodicals may also be consulted:
Polakov, Walter N., Kinetic Statistics as an Aid to Production and Distribution,
Journal of American Statistical Association, Sept., 1922, p. 359.
Clark, Wallace, Installing Gantt Production Methods, Industrial Management,
June, 1920.
The books of Mr. H. L. Gantt, Organizing for Work, Industrial Leadership, and
Work, Wages and Profit, also contain mention of the charts.
Chapter XXIV
SUMMARY CHARTS
Historical data often comes in three sets of figures showing
income (credits), outgo (debits), and balance. A wide variety
of names is used for these three sets. Sometimes they are
Production Imports
Consumption Exports
Fig. 243. The Flow of Goods.
called production, consumption, and stocks. Sometimes they
are known as export, import, and foreign trade balance.
Sometimes there are several groups of these three linked to-
gether in a single chain of events. Thus in a single business
concern, the purchasing agent will keep a record of his orders
(credits), receipts (debits), and balance on order. He will also
probably keep a record of his receipts (credits), uses (debits),
and balance on hand of raw materials. The factory manager
may keep a record of withdrawals from raw material stock
(uses) (credits), production (debits), and goods in process
(balance). He will certainly keep a record of production
(credits), shipments (debits) and finished stock on hand. If
goods are stocked in warehouses, the warehouse clerk will
keep a record of receipts (shipments in) (credits), sales or
consignments (shipments out) (debits), and stock on hand.
278
sOmmary charts
279
The accounting department will keep track of sales (credits),
payments received (debits) and accounts payable (balance
due). Other steps may be inserted between the above. In
these chain periods of groups of figures, the outgo (debit) from
one deposit station will appear as the income (credit) to an-
other.
In fact all historical data belongs to one or another of these
three classes. Population statistics are merely a balance be-
tween the number of births (income) and the number of
deaths (outgo) in a community or country. Crop and pro-
duction statistics generally belong to the income class, the
figures for consumption and goods in storage or process being
rarely available. Examples might be multiplied.
The economists’ distinction between stocks or funds of
goods and streams or flows of goods goes to the bottom of all
this. For if you will regard the stocks of goods as a reservoir
or body in repose, you will see that the income is a flow or
stream of goods into this reservoir and the outgo is a flow or
28 o
CHARTS AND GRAPHS
stream of goods out of this reservoir. And if you are math-
ematically inclined, you will notice that whenever any two of
these three sets of data are given us, we can easily compute the
third. If we know the January 1st inventory, the production
during the year, and December 30th inventory, we can easily
compute the shipments or sales for the year, that is, the
withdrawal or outgo.
Now there is a very important distinction between stocks
and streams (whether income or outgo streams). The former,
stocks, can only be measured at a point of time, while the
latter, streams, can only be measured during a period of time,
between two points of time. In the language of physicists,
streams of goods have one more dimension than stocks, for
they have the added dimension of time. And the result of all
this is that while you can cumulate or total up the stream
figures, you cannot cumulate or total up the stock figures.
You cannot cumulate daily the population of a city in order
to get the population monthly, for population is a stock
figure, though you could have cumulated the number of births
daily to get the number of births monthly. You cannot cumu-
late your monthly balance in order to get at your annual bal-
ance, but you must cumulate your monthly production in
order to get at your annual production. The distinction be-
tween stock figures and income or outgo figures is funda-
mental.
The usual method of showing these three sets of figures is
to use three curves upon a single chart. The use of three
curves for such dissimilar data is always confusing to the
reader of the chart, as the thin plotted line of the curve repre-
sents in one case a static or stationary stock of goods and in
another a series of separate and distinct additions or sub-
tractions. If the chart shows monthly data, a recasting of the
chart on an annual basis would raise the production and con-
sumption curves to twelve times as high a level on the chart,
but would leave the stock or balance curve unalFected.
The author has designed what, for the lack of a better name,
he is accustomed to call a summary chart to show these three
sets of figures together upon a single chart.
This summary chart is a combination of two vertical-bar
charts, and a curve chart in which the bars show the stream
figures (income and outgo) and the curve shows the balance
or stock figures. This distinction makes clear to the most
SUMMARY CHARTS
281
THE ACCmWLATED TRADE BALaBCE OP THE 0. S,
BBiliuit*d ♦xportB, importa and accuouXated (alnca 1800, trada balftno*
tint ted States, 1800-1920
(Hote:* Approximtiens xa paranthaeeB; all data aa of Jan, let)
9uitulated ® 8 8
balances 't
(I- ,000,000)
0» H «0
(O lO (O
C»» »- U>
w t- 0*
casual reader that the total income is made up of all the little
income bars and the total outgo is made up of all the little
outgo bars while the stock of goods changes according to the
fluctuations of the curve. In practice it has been found that
the chart succeeds admirably in showing simply and clearly
the rather complicated relations of the three sets’ of figures.
And the chart is more or less unique in its nice use of both
bars and curves simultaneously.
A color distinction is made between the income and outgo
bars, income being black or green and outgo red, or income
being white and outgo being black. A pale tint, gray or half-
tone, is given to the entire area between the curve and the
zero line, except where the bars cross this area. The stock
curve or balance curve being plotted at the beginning and end
28a
CHARTS AND GRAPHS
I
o 0-2
u
£
SUMMARY CHARTS
283
of each period of time, since these are the dates of inventories,
the plotting points for the curve are the ordinates of the various
points of time. The vertical bars on the other hand are placed
within the spaces for the periods of time (days, weeks, months
or years) between the ordinates. Two bars appear in each
space, the first for income and the second for outgo. These
bars should not be any wider than is necessary to make them
quite clear. The remaining space between bars is used for the
plotting of the curve and for the shading of the area under
the curve. In its finished form the curve, or stock figure, of
balance on hand, which might be called a band curve because
of its shaded area, forms the background of the chart. Against
this background, the quantity added to the stock each period
of time, and the quantity subtracted therefrom, that is, the
bars or stream figures, of income and outgo appear in the fore-
ground as solid bars.
The vertical scale for the summary chart (like the horizontal
scale) should be the same for all three sets of figures. When
this uniform scale is used it will be easily seen that the changes
or fluctuations of the stock curve exactly coincide in vertical
distance with the difference between the lengths of the two
stream bars. If the income bar is higher than the outgo bar,
the curve of stock on hand will rise by the difference in height
or surplus, and vice versa, if the outgo bar is the larger, the
curve will fall by the difference or deficit. Another advantage
of the uniform scale is that the height of the stock curve can
be compared with the height of the various outgo bars to show
easily the approximate period of time the stock on hand would
suffice if income were to cease. Where the stock represents
invested capital, a firm desires to keep this margin of stock on
hand as low as possible, and the chart shows clearly the size
of the inventory compared with the periodic requirements. In
the case of non-perishable goods or semi-perishable goods,
such as fresh food stuffs, the storage, or amounts withdrawn
from circulation, vary a great deal seasonally, and warehousing
conditions must be sufficient to meet the maximum stock
which will be in storage.
The summary chart can be made to carry a great deal of
detail by converting the bars into compound or segmented
bars, and the curve into a band-chart or segmented curve.
In this case, care must be used in the shading of the segments
of the bars, in order that they will not obscure the primary
284
CHARTS AND GRAPHS
distinction between income and outgo bars. It is, however,
possible to make all the shadings of the black or green income
bar in various degrees of intensity by various black or green
cross-hatched patterns, and to accomplish the same for the
segments or layers of the red outgo bar and the green or black
balance curve. The segments would indicate the various parts
of the income, outgo or balance. When much detail is shown
in this way, it becomes necessary to omit a portion of the data
PWCE CALL
offiTOCKS J^ATE
100
^0
60
4.0
£0
0-
«
10
8
6
4
• i
: f -
*. AVERAGE PWC
Scf >T6
J V*
AVERA(5E'*^i
CALLU0AN*»^n-,
RATE
A
\
STOCK T
RANSACTIONS
II
HILLIONS
of shares
30-
1
lilUl
III II mil
1
10 -
919
19^0
19^1
Fig. 247. A Customary and Sound Combination of Bars and Curves.
Average price of stocks, average call loan rate, and total transactions in stocks
each month . — Permission of Mr. Carl Snyder.
from the chart, or to adopt such wide intervals between ordi-
nates that the data can be entered horizontally. The latter
method usually calls for a chart of more than usual size.
The successful use in this chart of a distinction between
vertical bars and curves suggests that a general rule could be
made for all cases of complicated charts, wherever more sim-
plicity is desired. This rule would be that bars should be
used for streams of goods, and curves should be used for stock
figures. The rule would have many exceptions owing to the
convenience of the curve form in general, but it would appear
to be a sound principle to follow, wherever the choice is open
to us and the use of either curves or bars alone is not felt to
give sufficient clearness.
Chapter XXV
SILHOUETTE BAR-CHARTS
For certain purposes, only a few details of historical data
are required for each one of a large number of different and
heterogeneous phenomena. In price movements or stock
quotations, for example, the practical man is interested
OASOLIOT! STOCKS
Relative Figures of Stocks on Han*3 of Gasoline at End of Each Month
United States
1920-1921
(Average month, 1919 « 100)
(Sotirce:- U. S, Bureau of Mines)
86
COCRSE OP PnoCtTCTION SiNCB 1919.
EELATIVE PRODUCTION (1919-100).
Maxi-
mum
since
end of
1919.
Mini-
mum
since
end of
1919.
1920
aver-
age.
1921
aver-
age.
Feb.,
1921.
Mar.,
1921.
Feb.,
1922.
Mar.,
1922.
Foodstuffs:
Wheat flour...
125
64
82
91
* 64
82
! 88
89
Beef products
109
67
92
83
67
83
75
Porlcproducts
151
58
93
97
114
92
102
Lamb and mutton
110
58
80
94
89
102
1 70
Sugar (meltings)
165
40
104
92
80
133
128
165
Oleomargarine *
126
26'
103
60
70
73
1 42
52‘
Cottonseed oil.....
349
7
100
166
247
229
140
no
121
20
76
71
34
58
177
64
99
118
76
91
{
Cheese ...
. 169
41
86
83
49
68
Ice cream
468
42
111
153
44
71
Clothing:
Cotton (consumption). . . .
114
57
109
79
76
84
91
100
Wool (consumption)
126
42
83
95
64
83
111
124
_ Sole leather
95
63
82
79
63
72
78
78
Fuels: ^ ,
Anthracite coal
119
63
i 101
99
105
101
; 92
119
Bituminous coal
137
74
121
89
81
79
107
131
Beehive coke
127
11
110
30
55
36
! 35
46
By-product coke
62
122
* 79
90
85
1 86
102
Cmcie petroleum
i49
104
117
124
112
130
130
149
141
98
123
130
118
127
121
110
71
99
1 83
84
87
86
136
93
116
' 127
115
119
120
135
89
124
104
103
103
98
Electric power
119
98
113
105
98
105
107
117
Metals:
Pig iron
132
34
119
54
76
63
64
80
Steel ingots 1
140
34
121
59
74
66
74
100
Copper
83
17
94
37
71
83
35
58
Zinc
126
38j
105
47
46
41
69
Silver
129
$0
100
95
116
129
82
89
Gold
181
79
88
113
93
100
94
99
Tobacco: i
Cigars*
128
75
112
96
84
95
76
90
Cigarettes*
116
64
84
96
93
101
71
92
Manufactured tobacco * . . .
119
50
94
91
86 i
100
92
108
Lumber.
Yellow pine..
113
69
94
99
88
101
98
113
W'estern pine
121
20
121 1
67
20
67
38
53
North Carolma pme
153
33
98
88
63
71
149
153
California white and sugar
... 1
pme
204
8
121
78
n
12
19
15
California redwood
156
57
122 i
109
92 i
120 1
90
135
Pouglasfir
118
44
102
79
57 i
68
108
107
Idichigan hardwood
111
32
86
60
68 1
86 1
49
49
Northern hardwoods
161
21
105
88
117
147 i
72
118
Hemlock
120
33
91
57
67
52
44
67
Oak floorm&
202
42
106
123 j
55
84
171
202
Paper:
Newsprint
114
69
no i
89
90
94
85
103
All other paper
132
69
121
86
76
83
101
119
Mechanical wood pulp —
143
55
109 i
87 i
98
118
82
119
Chemical wood pulp
138
64
117 I
79 !
78
74
90
106
Corrugated paper board > .
129
30
104
65
42
49
86
100
Sohd hber paper board .
142
18
104 1
89 !
53 !
75 i
100
lie
Stone, clay, and sand prod-
i
ucts:
Sihca brick.,..
130
13
106
40
66
63
47
65
Clay fire brick
127
43
120
63
81 1
83
68 1
84
Face brick
121.
34
100
100
34 !
41
51
93
Cement
157
61
125
122
65
101
64
100
Class battles ...
124
48
104
69
87
6$
81
BtHLDiNG equipment:
1=0 I
Baths, enamel.
189
65
149 1
120 i
71
78
152
189
Lav atories , enamel
199
86
112
127
136
1 129 i 154 i
199
Sinks, enamel
170
80
no
122
96
128
135
166
Buildings (contracted for)
118
30
72 i
70
36
58
j 65
112
Transportation vehicles:
1
Automobiles, passenger...
1121
»ol
114
93
79
111
152
132
102
46
49
74
Locomotives.
135
13
89
50
79*
72
20
17
Ships
79
(*i
67
30
32
42
11
1
2
^ Since July 1, 1921 .
2 As represented by tax-paid withdrawals.
3 Eelative to last 6 months of 1919.
From Monthly Survey of Current Business.
Fig. 249» The Ees^ntial Data,
SILHOUETTE BAR^CHARTS
287
primarily in the latest quotation, but would also like to know
whether this last quotation is an increase or a decline from the
quotations on previous dates, and how it compares with past
maximum and minimum prices. Here, then, are four points
of interest to him, the present, the immediate past, and the
prior record-making peak and valley prices (regardless of the
dates or time of the latter). And the point is that we want to
see these facts, not for one only but for a large number of
commodities. It would be easy enough to present curves for
the individual commodities, or even to combine a few on one
curve-chart, but how can we present only the facts wanted,
for all the commodities, graphically in one simple chart ?
If you were to stand a historical curve-chart up on edge
and view it from the side toward which the curve is moving,
you might succeed in imagining that the curve was really
snaking its way directly at you. And if it had actual volume,
instead of being a thin line of ink, you would see most clearly
its nearest end representing the last value, and behind that a
short portion of its previous values, and still further back you
could make out the silhouette of its extreme peak and valley
points. Eureka! These things are all you wanted to know
about each individual curve, and seen from the planes in
which the curves lie, each curve compresses to the width of its
imaginary columns or vertical bars. This suggests the method of
graphing to which, for lack of a better one, we give the name of
silhouetting. The silhouette curve-bar is a recent development
in graphics and probably has not yet reached its final stage.
Since the graph is really a projection of a large number of
curves shooting straight out of the page toward the reader,
something must obviously be done to lift the nearest ends of
the curves from their other and earlier positions. The method
of segmented bars alone gives too much flatness to a picture
which is really a projection of three dimensions on two — a con-
densation of a three dimension model into a two dimension
sheet. For this reason it is obvious that the nearest ends of
the chart stand out clearly and appear to be wider, precisely
as if photographed from a real model. The eflFort here being
to produce the effects of perspective, the portion of each bar
which represents its latest reading or value should be of full
width, but the earlier readings, and in particular the past
peaks and valleys, should be considerably narrower, to give
the effect of greater distance.
288
CHARTS AND GRAPHS
If the portion of each bar connecting the latest and the
next previous values be kept of uniform width, it is necessary
to show whether the change has been one of rise or fall, that
OASOLISE STOCKS
Relative Plgurea of Stoclra on Hand of Gasoline
at End of Each Month
United States
1920-1921
(Average month, 1919 * 100)
(Source:- U* S. Bureau of Mines)
Fig. 250. The Same Curve Seen From Its End.
is, which end of the bar is the latest reading. This can be
indicated with a small arrow-head in the bar, or by solid
shading of one color for rises and of a totally different color
for declines, or by both methods together. Moreover the
latest reading might be indicated by a star or other symbol
which the reader can quickly glimpse. On the other hand, if
COMMODITY STOCKS
Figures of Stocks of Specified Commodities
United States
Miirch, 1922
(1919 Average month • 100)
(Source.- Survey of Current Business)
y.,y
y
I
KEY
March 1920
Maximum since 1919
Minimum since 1919
Fie. 251.
“'n u.
->T r
A Detailed Form.
Pebruai*y I9i..
March 1921
290
CHARTS AND GRAPHS
we are willing to make the bar of tapering shape, then the
narrower end could indicate the earlier, and the wider end,
the later value.
A more pictorial method may, however, in some cases
prove to be the more efficient means of flashing the story of
the chart to the reader. Thus, if a large circle be used for the
latest value, and a small circle for the next previous, with a
triangle or star for the highest past peak and an inverted
KETt
j" * " ■ • " j Average price in 1921
Average price In 1920
I i
r *1 Average price In other year*
Majtlnum price (since 1912) and date
f •** Minimus price (sinoe 1912) and dat*
KETAll, PRICES
Relative Figures of Annual Average Prices at Retail of Speclfiaa Coraraoditlet
United States
1917-1921
(1915 average - 100)
(Source - United States bureau of Labor Statistic#)
Fig, 252. Data in the Chart.
SILHOUETTE BAR^CHARTS
291
triangle or square for the lowest past valley, it would seem
that the results would be easily understood at a glance. Each
of the four values could be strung upon the same central line
(or ordinate) serving as a connecting thread in the place of
the bar. The two circles for recent values could be white for
rising values and black for declining ones. Such a pictorial
system would make possible the addition of even further
symbols, such as still smaller circles for the second previous
reading when this was sufficiently different from the last two
readings, and smaller triangles for minor peaks and valleys.
The peak and valley symbols could contain numerals repre-
senting their years or approximate dates. The enterprising
reader will find still other embellishments, which, so long as
they increase the ‘Visibility’^ of the facts or the speed with
which they are flashed to the accustomed and unaccustomed
eyes, will be justified.
The labelling of the various bar-curves in this chart calls
for great care, particularly when the data is heterogeneous.
Each compressed curve should be so distinctly and clearly
NORMAL
100 PER CENT.
\
j MARCH \QIZ
V/OOL CONSUMPTION
WHEAT FLOUR. |
PETROLEUM
ANTHRACITE COAL
CEMENT
,T1N
MEAT5UUGHTERED
LUMBER
WOOD PULP
BITUMINOUS COAL ;
PAPER !
TOBACCO
COTTON CONSUMPTION 1
STEEL INGOTS^ i
riG IRON j
Fig. 253. Simple Silhouette Bars Presented Horizontally.
Production of basic commodities in March 1922, and the low point in 1921
compared with normal production. In cases in which March production figures
are not available, February figures are shown — Ptn mission of Mr. Carl Snyder.
192 . 1 *
LOW
SUGAR meltings
COMPAMSON OP PRESENT WHOLESALE PRICES WITH 1920 AND PRE-WAR.
(Relative prices 1913* XOO.)
index NUMsens
WHEAT
CORN
potatoes
COTTON
COTTON SEED
WOOL
cattle. BEEP
HOCS
LAMBS
wheat, spring
WHEAT, WINTER
CORN, NO 3
OATS
BARLEY
hVE. NO 2
TOBACCO- BURLEY
COTTON. MIDDLING
WOOL. OHIO. UNWASHED
CATTLE. STEERS
HOGS. HEAVY
SHEEP. EWES
SHEEP. LAMBS
FLOUR, SPRING
FLOUR. WINTER
SUGAR. RAW
SUGAR. GRANULATED
COTTONSEED OIL
BEEF. CARCASS
BEEF. STEER. ROUNDS
PORK. LOINS
COTTON YARN
COTTON PRINT CLOTH
COTTON SHEETING
WORSTED YARN
WOMEN S DRESS GOODS
suitings
SILK. PAW
HIDES, PACKER'S
HIDES. CALFSKINS
leather, sole
LEATHER. CHROME
BOOTS AND SHOES
COAL, BITUMINOUS
COAL, ANTHRACITE
COKE
PETROLEUM
PIG IRON, FOUNDRY
PIG IRON. BESSEMER
STEEL billets
COPPER
LEAD
TIN
ZINC
Lumber, pine, southern
LUMBER. DOUGLAS FIR
BRICK. COMMON. NEW YORH
BRICK, COMMON, CHICAGO,
CEMENT
STEEL BEAMS
RUBBER. CRUDE
SULPHURIC ACID
From the Survey of Current Business.
Fig. 254. A Silhouette-Bar Chart Set Horizontally.
SILHOUETTE BAR^CHARTS
293
labelled by its descriptive title at the base of the chart im-
mediately underneath it, that the reader may have no difficulty
in finding the graph for any particular item in the series, or
in finding the title of any particular graph. Because these
charts generally contain a very long list of items, it is well to
divide them into smaller groups with slight margins bet’ween
the groups, and it goes without saying that these groups
should be as logical and useful to the reader as possible.
The silhouette curve-bar can be projected on either an
amount-of-change or a rate-of-change scale. Usually the
index numbers are used in the place of the absolute numbers,
so as to make the items and graphs comparable, and usually
the scales are arithmetical. But index numbers can equally
well be shown on logarithmic scales and the latter give an
added refinement to the chart which shows with greater
accuracy the significance of changes, to those readers who
understand it.
Chapter XXVI
INDEX NUMBERS
It is one of the most important functions of the statis-
tician to compare the behavior of different phenomena and
find out whether there appears to be a relation of cause and
effect between them. For if such a relation exists, we ordin-
arily expect to find evidence of it in their fluctuation. If one
phenomenon is directly or indirectly the cause of the other
we may expect to find the fluctuations of the first paralleled
or mirrored in the fluctuations of the second. If both are the
effect of a third common cause, we may still expect to find a
marked similarity between their movements. Often there is
a delay or lag between the time of the movements of one object
and the reaction upon the movements of the other. Thus it
has been shown that the fluctuations in the production of pig-
iron follow closely those in the corn crop after a lag of about
two years.
The science of business forecasting is largely built up on
these comparisons. Thus if pig-iron activity follow the corn
crop exactly after a two years lag, it is easy to see that with
a knowledge of corn prices today we would be able to forecast
the prices of pig-iron two years from today. Some of the
ablest statisticians in the country are engaged upon research
in this forecasting problem. Unfortunately the relations are
not simple and easily established, and the available informa-
tion is not nearly complete enough at present to make general
business forecasting very successful. It is, however, sometiriies
possible for a capable mathematician to construct a very
accurate forecast for an individual business or industry in
which the determining factors are more easily ascertained and
measured.
When a condition of similar fluctuations ^either parallel or
mirrored) exists, the word ‘^correlation^' is used for the condi-
tion by both mathematicians and statisticians. They have a
194
INDEX NUMBERS
295
complicated, mathematical process or formula for computing
the degree of this correlation between two or more series of
figures. The degree or extent of this correlation they indicate
by a “correlation co-ejfficient.’’ The mathematical work in-
volved in determining this correlation co-efficient, which
measures the degree of correlation between two series, is long
and complicated. There is, however, a very simple method of
detecting the existence of correlation or similarity of fluctua-
tion between two series, which consists of plotting the curves
of the two series and comparing the curves by sight. Do the
wiggles in one curve parallel or mirror the wiggles in the other?
If the curve has been properly plotted, correlation provided it
exists, will be apparent at a glance. By using very trans-
lucent paper, you can subject the two curves to “light an-
alysis,’’ that is, you can lay one curve over the other, hold
the two of them up to the light, and immediately see the
slightest deviation of one from the other. The method does
not give the correlation co-efficient, or exact measure of the
degree of correlation, but it serves to give a sufficient idea of
the amount of correlation for most purposes.
The only problem is to plot the two curves correctly.
Ordinarily, the different series of data have different units
of measurement. Thus corn is measured in bushels and pig-
iron in tons, one a measure of volume and the other of weight.
So far as prices are concerned, they read in common units of
value, but one may lie much higher up on the scale than the
other curve, because one may be measured in dollars and the
other in cents. And as you know, the same fluctuation will
be greatly magnified in a curve lying higher up on the chart,
than in one positioned low. The fluctuations might be identical
and yet when plotted on the same scale of numbers they
would appear very dissimilar, because of the exaggeration of
the fluctuations in the curve of iron, lying higher upon the
chart.
There is however a very simple method of reducing two
entirely different series to a common scale with a similar posi-
tion and range on the chart. The trick is to use “index figures."”
In a previous chapter, the distinction between absolute and
relative figures has been pointed out. The bushels of corn are
absolute figures, but the percentages which these figures are
of figures at a certain point of time are relative figures, also
called index figures, index numbers and indices. When using
296
CHARTS AND GRAPHS
indices for a historical series, it is necessary always to state
what year or time is considered as the basis for the percentages,
that is, which year or time is taken as one hundred per cent.
The hundred-per-cent year is called the base-year or ^Tase”
and the price or value at this time is called the ‘Tase figure’’
for the relative series or index figures.
The first step in reducing a series of data to index figures,
therefore, is to select the base. Obviously a great deal de-
pends upon the base you choose. If you select the highest
P£f? CENT
Fig. 255.
Sales of 57 department stores in the Second Federal Reserve District and 8
leading chain stores doing a country-wide business (average monthly sales in
1919 — 100%). Permission of Mr, Carl Snyder.
figure in the series the rest of the series will lie below the 100%
line on the chart; if you select the lowest figure in the series,
the entire curve will lie above the' 100% line. The common
practice is to use an average of a number of figures during
times which were considered normal. Thus in its ^^price rela-
tives/’ or index figure of prices, the Bureau of Labor Statistics
has adopted the average price for the year 1913 as the base
in all its price-series on the general theory that this was the
last normal pre-war period of time. Others have adopted
other periods of time as the bases for their series. In special
cases you may have to adopt a special year regardless of its nor-
CHARTS AND GRAPHS
•298
malcy. In comparing the index figures from different sources,
you must convert both series to a common base year. A rela-
tive series can be changed from one base to another in the
same way that it was constructed from the absolute series,
that is, by dividing the series through by the value for the new
base period.
m&WCncM dP liAlTOFACttJRKD GOODS
Hiyaioal Voltaad of Produo-^ion and Growth of Population
United States, 1899-1919. CSourco;- from Mr. E. E. Day)
^0!^C>O)<T>C>OiOO>OO>O>C>O>Q>€^O)iOO1k<|9k4Pli
Fig. 257, Obviously, only Index Numbers are Possible.
Many books have been written on the subject of index
numbers .1 There is nothing difficult about the task of con-
structing relative figures for a single series of data. As already
stated, you first select a base for the series and then divide
^ The literature on this subject is considerable. In particular, the student should
refer to the works of Wesley C. Mitchell; also to Irving Fisher, ‘'Best Form of Index
Numbers/’ American Statistical Association Quarterly, March, 1921, p. 533.
JNDKY NUMBERS
299
all the other figures of the series through by this base figure,
turning them into percentages of the base, that is, into a rela-
tive series. To compare two series on a chart, you merely
turn them both into relatives to the same base, then plot the
curves of the two series and compare their fluctuations. The
difficulty comes when you want to make a common index series
for two relative series. Thus, if you have the price-series of
various grades of steel, how will you make a single index series
of figures for steel of all grades, that is, how will you combine
these various relative series into one single index series. For
if you are going to compare steel and wheat prices, it is obvi-
ously not an easy matter to have to compare a thousand dif-
ferent grades or kinds of steel with as many different grades of
wheat. It is much easier if you have a single index figure for
the price changes of all steel, and another for the price changes
of all wheat. The problem of finding a series of figures which
PERCENT
Fig. 258. Various Indices of the Same Phenomenon Produced by
Different Methods of Weighting.
Index of the prices of 20 basic commodities compared with the Department of
Labor Index (325 commodities ). — Permission of Mr, Carl Snyder.
300
CHARTS AND GRAPHS
will serve as index numbers for several relative series is not an
easy one.
Briefly, an index for a number of relative figures must be
some sort of an average of those figures. But there are several
kinds of averages, each with its own particular merits and pur-
poses. For simplicity, let us suppose that we have only two
original series, the price of a loaf of bread in the city of Osh-
kosh, arid the price of a loaf of bread in Kalamazoo, and wish
to find a single index for the price of a loaf of bread throughout
the county containing these two towns (assuming that they
together comprise the total population of one county). If
PRICES OP OIL STOCK AND< PETROLEUM
Rdlative prices of 20 oil iharee, petroleum producti,
and crude petrolexna*
( 100 ?{ • . 1919 )
(Sourcet- Po^ue, Bconomica of Petroletm)
Crude
Petroleum
pro3uot«
8 S S S
f-% #-«
Fig. 259.
INDEX NUMBERS
301
Kalamazoo bread sells at 10c a loaf and Oshkosh bread at 15c,
the average price would appear to be 12^c. This is the simple
arithmetic mean . of 10 and ISc. If we are using a geometric
mean as an average, the average would be around 12c, and if
we are using a harmonic mean as the average, the average
would be around 13c. For most purposes, however, the arith-
metic mean, that is, the common or garden variety of average
will do.
But let us suppose that Kalamazoo has a population of 900
persons and Oshkosh only 100 persons. Hence, for every loaf
eaten in Oshkosh there will be nine loaves eaten in Kalamazoo
and the true average price of every ten loaves will be about
\Oy 2 C. A little study will show that the average loaf is nine
times a Kalamazoo 10c loaf for every single time it is an
Oshkosh 15c loaf. In other words, we must “weight” the fig-
ures befoi'e averaging them. Now this weighting is sometimes
a very difficult problem. How would you combine changes in
the cost of butter and changes in the cost of other foodstuffs
UDES ilBUS
«f limrlj itmimu i)i CItII imd Wetld t«r«
United 1«60*7» «ad 191S-4I
(Souroe:- Monthly lAbor Effiifc)
Fig. 260 ,
CHARTS AHD GRAPHS
302
to get a common index, an average change in the cost of all
foodstuffs, which could be used as a cost-of-living index figure.
fAcis, i^icie affUiKa.®
INBEX number:^
304
CHARTS AND GRAPHS
In this writing, nothing more can be done than to indicate the
problem. It has not yet been finally answered.
As to the plotting and other details of chart-making for
index numbers, the rules laid down for historical charts in gen-
eral apply. The use of index numbers, however, generally
brings all data to a common scale and range of variation so
that a uniform charting field should be used for these charts,
when you are preparing a series of them. The uniformity is of
Fig, 263 . Correlation Shown by Mirroring.
The chart shows Foreign Exchanges on New York below the heavy line (m terms
of depreciation from parity), and commodity prices above the heavy line. —
Permission of Mr. Carl Snyder,
INDEX NUMBERS
305
value for making comparisons. It is well to place the data of
the original or absolute series beside the data of index or rela-
tive figures in the data attached to the chart, whenever the
relative has been computed directly from absolute data. Of
course this is not desirable where the indices have been com-
piled from a large number of original series.
PRICES
PERCENT.
LTABILITY
fMOUSAND^
OF DOLLARS
Fig. 264. A Slightly Lagged Correlation.
Average liability of failures in the United States each year compared with changes
in wholesale commodity prices. (Department of Labor index .) — Permission of
Mr. Carl Snyder.
Relative figures and index figures^ can be frequently used
for all sorts of data other than historical data, but the principles
and applications are the same and the greatest use occurs for
relatives and indices in historical series. They afford a simple
®The distinction between relative figures and index numbers is really very clear
and should be adhered to. Relative numbers are directly related to absolute data;
the absolute data is that original series of actual figures which has an individual as
well as a collective meaning. Thus, price-quotations are absolute figures. From these
absolute figures we derive relative figures by the process of division, using a constant
divisor which we call the “base-figure."’ But the relative figures, so derived, have no
individual significance; they take on a meaning only collectively, as a series, each being
a ratio between two absolute figures. Relative figures are but one step removed from
absolute figures.
Very different from these, are index numbers. These have no corresponding
absolute data; they are merely indicators of some theoretical and wholly imaginary
idea, such as the combined movement of many individual things. They are usually
derived from relative figures, as explained in the text, either by simple or by weighttsd
averaging.
3o6
CHARTS AND GRAPHS
and sound means of comparing any series, bringing widely dif-
ferent series, or even series measured in different units, together
into easily compared curves. They are at bottom no more
Fis:* 265*
Wholesale commodity prices in four countries (average prices in 1913 = 100%).
— Permission of Mr. Carl Snyder.
than percentages, though different from the percentages used
in 100% bars and band-charts, in that the value of 100% is
no longer a total, but merely one of the values in the series.
That the comparison of curves is largely incidental to the
search for correlation between the phenomena which the
curves represent, and that correlation studies are in historical
series very often directed to the practical end of forecasting or
predicting future conditions, are details which in no way limit
the general usefulness of indices. You may be seeking light on
the probable level of prices in your business in the future; this
calls for forecasting and therefore for a knowledge of attendant
and preceding developments. But you may also be only in-
terested in the relation between changes in your advertising
INDEX NUMBERS 307
appropriation and in your gross sales; in this case^ too, you
can well use index numbers or relatives.^
^ A very different type of relative figure is the ‘‘chain-percentage” or “link-relative.”
Being anti-logarithm of the logarithmic differential (or successive differences) of a
series, it has very little value and is rather over-estimated. It is secured by taking
each item as a percentage of the preceding item in a series (sometimes after subtracting
100), and has, therefore, no constant base. It is discussed in later chapters.
Chapter XXVII
FREQUENCY SERIES
We have now to consider the curves for data of a non-
historical nature, that is, data in which time is not the inde-
pendent variable. This is often data of conditions at a single
moment of time — a cross-section, as it is sometimes called, of
the phenomenon. At other times it is a compilation or recap-
itulation of phenomena (events or conditions) through a period
of time — still, if you please, a cross-section. The analysis of
such data proceeds through a series of changing conditions, and
the conditions can sometimes be so coherently arranged as to
form a variable. When this is the case, the data can generally
be profitably shown and studied by means of a curve-chart.
Curves of this nature are called ‘‘frequency curves,” a name
which is derived from their chief purpose, which is the display
of the frequency with which the phenomena occur under
given conditions. They are also sometimes called “picto-
grams,” but the latter name has fortunately not found general
acceptance.
A few examples of this type of data will serve to make the
class clear. The manager of a chain of retail stores, and to a
CITY FINANCES
Percaplta Revenue Receiuts and Cost Paymenta
United States
Year Ended June 30, 191R
(Source:- Tlnited States Census)
Population
of Cities
(1917)
Psroapita
Revenue
(Dollars)
Percapita
Expense
(Dollars)
30^000 - 60,000
27.14
26.23
50,000 - 100,000
26.23
27.29
100,000 - 300,000
29.18
32.10
300,000 - 500,000
39.53
40.18
000,000 and Over
41.87
40.78
Fig. 266.
308
FREQUENCY SERIES
309
lesser extent any distributor over a large territory, will be
benefited by a report showing the per capita sales in cities of
different sizes, as such statistics will show him the comparative
value of large and small town outlets for his goods. Here the
classification would be according to the population of towns,
and for certain purposes a simple analysis would show the
average per capita sales in towns of each size. Again a manu-
facturer is putting up his products in many different sizes and
(48 - can)
SI 25 CASES
1/2 pound cans 1*200,034
1 pound cans 13,901,692
1-1/2 pound cans 1,529
2 pound cans 3,00$
PRODUCTION OP RED SALMON
Output of Canned Red or Sockeye Salmon.
Alaska Fisheries
7-year Total, 1913-1919
(Soxttce:* "United States Bureau of Fisheries")
Fig* 267.1
cases, and an analysis of sales according to size might take
the form of a table showing sales of each size which again
might be charted in a frequency curve when the sizes form a
connected mathematical series. And to take one more example,
the manufacturer of building materials might find advan-
RENTS IN DENMARK
Average Yearly Rentals of Famxly Dwellings
banish Cities
1918 and 1919
(1 Crown at par * 26,8 cents)
(Source*- Monthly Labor Review)
Capital Provinces
1918 1919 1913 1919
— Crowns — )
1 room and kitchen 138 148 85 99
2 rooms and kitchen 290 304 179 198
3 rooms and kitchen 413 434 271 301
4 rooms and kitchen 544 566 379 420
5 rooms and kitchen 780 828 602 557
6 rooms and kitchen 1065 1117 632 706
7 rooms and kitchen 1369 1464 706 851
2103 2328 1033 1151
Fig, 268.
6 rooms and more
310 CHrlRTS AND GRAPHS
tageous a curve showing the number of one-story, two-story
and higher buildings in his territory.
Both the historical and the frequency series are numerical
distributions, that is, their independent variables are mathe-
matical series or progressions. When the independent variable
marks specific points or periods of time we call the numerical
distribution a historical series. In all other cases we call it
a frequency series. Nor is it possible to apply this distinction
always, for there is a large class of numerical distributions in
which time is counted — not from a single common origin-point
of time but from various and usually unrecorded and unim-
EPFECTS OP DIPHTHERIA ANTITOXIN
Chances of Recorery due to Use of Antitoxin
On Various Days after Diphtheria is Discovered
(Sourod:- Kolle and H«tach)
Day of Disease
on »hlch
Antitoxin
is firet
Adainiatered
Percentage
of Cases in
which
Recovery
is M&do
1
lOO
Z
96
a
86
4
77
a
61
«
46
Fig. 269.
portant reference-points, and these also are to be classed as
frequency series. Counts or classifications (i.e. distributions)
of the population by age in years, or of mortality-rates,
marriages, weights, heights, illiteracy, or the like, by ages;
of orders or shipments by length of time taken to complete,
or the like, are examples of frequency series which have as
their basis, time.
In the consideration of frequency series, we may regress a
moment to the general subject of statistical tabulation. For
it is in frequency series that the greatest measure of statistical
treatment is called for, not alone in the handling of the com-
pleted series, but in the preliminary work of compiling the
series. And in actual practise the student will encounter a
baffling heterogeneity of frequency distributions, presented by
their compilers in various shapes and statistical fashions and
often, unfortunately, in what he will come to recognize as
FIBE LOSSES IN THE OHITED STATES ^
Slatiotici of Loeees of Property due to Fire in Larcer Cities U
1919
(Sources- N&tiorval Board of Fire Underwriters)
Humber Property Lose
state
City
population
Fires
Total
Peroaplta
Ala
Bimunghaa
225,000
2,886
691,207
2.63
cal
Los Angeles
700,000
3,100
1,398,206
1«98
Oakland
226,000
1,603
136,746
.61
San FranOlsoo
626,000
3,351
Col
Denver
290,000
1,638
374,214
1.29
Conn
Bridgeport
200,000
762
118,499
.59
Hartford
140,000
661
215,695
1.64
New Haven
175,000
874
286,717
1.62
Waterbury
100,000
459
133,995
1.34
B . ff .
Washington
400,000
1,662
683,171
1.46
Fla
Jacksonville
110,000
423
250,298
2.28
Ga
Atlanta
230,000
762
665,336
2.68
111
Chicago
2,816,000
17,208
7,331,023
2^60
Ind
Indianapolia
300,000
2,969
1,068,937
3.56
la
Des Koinea
108,000
925
285,338
2.64
Kan
Kansas City
100,000
984
156,766
1.66
Ky
Louisville
265,000
919
662,204
2.08
La
New Orleans
380,000
882
548,248
1.44
Ud
Baltimore
760,000
3,244
3,206,602
4.27
Haas
Boston
808,310
4,934
2,677,584
3.19
Cambridge
112,000
684
418,363
3.73
Fall River
130,00 b
414
210,631
1.62
Lawrence
106,000
622
78,120
.74
Lowell
118,000
943
232 , 103
1.96
Lynn
104,000
631
96,099
.91
Hew Bedford
120,000
643
249,917
2.08
Springfield
130,000
812
367,947
2.76
Worcester
190,000
1,345
246,839
1.30
Uioh
Detroit
900,000
4,190
4,026,279
4.47
Grand Rapids
145,000
1,093
767,604
6.22
Uinn
Duluth
100,000
415
169,807
1.69
Minneapolis
400,000
2,279
924,733
2.31
St. Paul
276,000
1,299
633,140
2.30
lie
Kansas City
320,000
3,297
1,027,052
3.21
St. Louis
900,000
4,088
1,616,254
1.80
Neb
Omaha
205,000
1,233
293,446
1.43
N,J.
Camden
110,000
436
76,933
.70
Elitabeth
110,000
410
99,013
.98
Jersey City
300,000
1,169
413,563
1.38
Newark
450,000
1,645
896,881
1.99
Patterson
130,000
473
393,197
3.02
Trenton
108,000
403
287,079
2.66
N.Y.
New York City
j , 006, 794
13,429
12,488,258
2.08
Rochester
300,000
913
390,375
1,30
Schenectady
108,000
316
69,596
.66
Syracuse
160,000
567
263,527
1.68
Yonkere
106,000
482
168,779
1.69
Ohio
Akron
175,000
835
389,819
2.23
Cincinnati
418,022
1,468
612,742
1.47
Cleveland
750,000
3,906
1,793,044
2.39
Columbus
240,000
823
249,375
1.04
Dayton
153,930
1,081
300,361
1.95
Toledo
220,000
1,066
1,500,075
6.82
Youngstown
130,000
745
199,618
1.53
Okla
Olkahoma City
110,000
510
402,080
3.65
Ore
Portland
326,000
944
562,831
1.70
Pa
Erie
112,000
425
100,267
.89
Philadelphia
1,850,000
4,204
4,886,485
2.64
Pittsburgh
600,000
2,580
1,707,007
2.84
Reading
110,000
141
138,218
1.26
Scranton
150,000
405
502,811
3,35
P.I.
Providence
260,000
1,766
627,611
2.41
Tenn
Memphis
166,000
1,738
660,993
3.92
Nashville
160,000
667
406,751
2.71
5ox
Dallas
140,000
893
253,436
1.81
Fort Worth
110,000
637
226,938
2.06
Houston
160,000
1,003
1,010,062
6.73
San Antonio
160,000
401
203,996
1.36
Utah
Salt Lake City
130,000
672
347,066
2.67
Vm
Norfolk
160,000
773
4,084,267
27.23
Richmond
170,000
741
134,426
.79
Vaih
Seattle
380,000
2,358
762,757
2.01
Spokane
167,625
729
334,617
2.13
Tacoma
123,000
794
163,863
1.33
IILtO
Milwaukee
610,000
2,?28
S 17,336
1.70
Figr. 270, The Raw Material for a Frequency Series,
312
CHARTS AND GRAPHS
various stages of compilation. While we cannot attempt to
cover the subject as thoroughly as it is treated in the statistical
text-books, yet we may well give it a brief survey, in order
that the chart-maker may be enabled the better to construct
his frequency curve.
The first stage in the preparation of a frequency curve is
the simple listing or list of observations. This list is not in
any sense a frequency series; it is merely the crude form, the
raw material, from which the frequency series will be made.
The stubs of the list are names, or numbers, which can be
Check
Auerat^e
number
daily
of man
output
4003
381
4182
380
4206
370
4215
392
4220
400
4221
414
4223
894
4200
413
4232
416
4238
807
4276
892
4282
374
4287
347
4289
406
4842
377
4860
428
4864
890
4856
898
4361
402
4370
387
4373
382
4892
411
4395
408
4818
410
4402
391
4419
407
4426
425.
4462
399
4465
401
OUTPUT OP 10RKH.RS
(Hon-«tereotyp 0 d
opdration)
(Source:- Report
of P* S* Florenco)
Fig. 271, Another Crude List.
called items, and the list itself is composed of numerical values
or other observations which have been noted for these items,
FREQUENCY SERIES 3 1 3
It is possible for both stubs and observations to take the form
of numbers, but still the list does not form a series. Also it
is possible for both stubs and observations to be abstract,
that is, not numerical. Usually the stubs are not numerical,
while the observations are. The length of the list, that is, the
number of items in it, indicates the total ‘^population’^ or
‘"universe’" of the distribution or series which will be formed.
A universe of much less than a hundred items is not likely to
prove a very reliable “sampling;” as a rule the sampling should
be considerably larger and detailed reliability can generally
be had only in samplings which contain thousands of observa-
tions. The trustworthiness of a sampling also depends, of
course, on the size of the external or unobserved universe, as
well as upon bias and error in the selection of observed items
or making of observations.
The second stage, that is, the first step in the conversion
of this list into a series, is the rearrangement of the list in the
order of magnitude of observations. This is done to facilitate
Bridgeport
tO.69
Fall River
1.62
Naahville
2.71
Ooldond
.ei
Oultttb
1. 69
Springfield, Me
.2. 75
Soheaeotidf *
.66
Portland, Ore.
1,70
Pittsburg
2.84
Ceadera
.70
Vilwaukee
1.70
Atlanta
2.85
Lavronoo
.74
St. Louis
1.80
Fatteraoa
8.02
Rlobnoad
.79
Dalles
1.61
Boston
3.19
Brie
.89
Day too
1.95
Kansas City,llo<
.8.21
Lpna
*91
Losell
1.96
SorantOB
3.36
Blisabetb
.98
Los Angelas
1.98
Indianapolis
3.56
ColttBbue
1.04
Hssark
1.99
OklahoBS City
3.65
Readiag
1.26
Seattle
2.01
Caabridge
3.73
Dearer
1.29
fort forth
2.06
Uenphla
3.92
•eroeeter
1.30
Loulsvills
2.08
Baltinore
4.27
Roobeetar
1.80
Rev Bedford
2.08
Detroit
4.47
laeoao
1.23
Rev York City 2.08
Grand Rapids
5.22
Vaterbury
1,34
Spokane
2. 13
Houston
6.73
Saa baton lo
1.36
Akron
2.23
Toledo
6.82
Jereey Cit|
1.38
Jeokaonrllle
2. 28
Horfolk
27.23
Oaaba
1.43
St. Fata
2.30
Rea Orlaaae
1.44
liiaaeaporlla
2.31
Vaablagton
1.46
Clorelan^
2.89
ClaolnBatt
1.47
Frovldenoa
2.41
Me* Haven
1*52
Chloago
2.60
Yottagatowo
1.53
Blrniogban
2.63
Hartlord
1.54
Dea Moines
2,64
Kansaa City
1.55
Fblledolpbia
2.64
Syraouio
1.58
treaton
2*66
yonke re
1.59
Salt Laka City 2. 67
PSRCAfltA riR'S LOSSES
lit 74 Iftrg* Aaorioas oLtl«»
1919
Fig. 272. The First Step is Arrangement by Magnitude.
a count, which will shortly take place, or to enable us to sum
up two or more sets of observations about the same items in
the list. Notice that we now arrange by the observations and
not by the stubs or items. Already the emphasis has shifted
CHARTS AND GRAPHS
3H
to what in a broad sense might be called in this crude list a
dependent variable. The reason is that we are about to forget
the stubs altogether and make the observations (which occupy
the place of a dependent variable) the independent variable of
our series. In other words we are about to distribute the data
according to the numerical size of its parts.
The third stage, and the step which finally yields us a
frequency series, is to gather the observations into groups, or
1 10, 000
e,ooo
«,250 (2)
«.000 ( 8 )
5.760 (3)
5,500 (32)
6.860 ill )
6,000 ( 82 )
4.860 (8)
4,800 ( a )
4.760 (12)
4,600 (2)
4.600 ( S 6)
4,400 (16)
4,260 (1«)
4.800 (8)
4,160 (16)
4,000 (207)
8, S 87
8.600 (2)
8,860 (2)
8.800 (7)
8,781 (2)
8,780 ( 49 )
8,700 (8)
8,660 (2)
8,400 (108)
8,800 (1 X 7)
8,460 (11)
8.487 (2)
8.400 (5)
8,860 (3)
8,300 (69)
8,260 (64)
3,200 (36)
8,160 (6)
8,100 ( 27)
8,060
8,050
8,000 (819)
8,900 (16)
8,870 (8)
8,860 (10)
8,620 (2)
2,800 (74)
2,780 (68)
2,700 (120)
8,690 (4)
8,660 (20)
8,640 (8)
8,600 (101)
8,670
8,620
3,600 (197)
8.460 (2)
2.460
2,420 (2)
2.400 (91)
8,860
8,840 (8)
2.820
8«800 (39)
2,276 (8)
8,270 ( S )
2,250 (22)
2,220
2,200 (87)
2,100 (52)
2.000 (91)
1,900 (9)
1.600 (46)
1,760 (10)
1,700 (16)
1,660 (8)
1,600 (8)
1.600 (10)
1,467
1.400
1,860 (4)
1.000 ( 2 )
600
800
CQLISQS 2B0TBS80R3< SlLARlIS
Sklftrle* p»id to full protesoors la tho
oolleges fta4 aalTsrsitlos (pakllo tactl*
iutlnns onlp), Oaltod States, 1020.
(Souro«:« O.S.Bareaa of Idaoatloa.)
(Total ottober • 8,460)
Fig. 273. A Common Tendency to Bunch Up.
classes, and record the count of the number of observations in
each class. Now the observations have become the stubs and
are the independent variable, while a new dependent variable
has been created by the counts of the observations of each
magnitude (that is, the number of observations occurring in
each class). Here we have clearly an arbitrary choice of the
independent variable. Had we recorded a different feature of
the same original items in our crude list, we should have had
a diflFerent independent variable for our final frequency dis-
tribution. Had we recorded two sets of observations for each
item we should have had to make our choice between two
FREQUENCY SERIES
315
1920
(Souro* - Onltad Ststas Bureau of Edusatlon)
(Total tiuaber s 2,460)
CHotat* Beale ihovi amount of ealarj in dollare, linae aho« numbor of profaeaore raoaiTln|[ eeae.)
Fig:* 274. Piling Up on the Round Numbers.
possible independent variables for the same series. When
these alternative possibilities are presented, a wide variety of
resulting series may be formed. The final dependent variable
may be a count (which forms the frequency series in the strict
sense) or may take the form of rates, ratios, percentages, or
averages (which are only in a general sense called frequency
series). It is not our purpose here to make an exhaustive
study of these possible varieties and combinations; we present
the reader only with a brief explanation of the simple frequency
series (strictly so-called) in which the dependent variable is a
count of the number of items falling within the class or group
limits.
It may seem at first tnought a very simple proceeding to
gather the items into classes or groups, as above described, but
the fact is that at this point much statistical skill is called for.i
For the size of the groups will determine their number, and foi
the best results graphically, there should be from fifteen to
twenty groups. But we must not only strive for a sufficient
number of groups, but we must also consider the precise loca-
tion of their limits. The limits of the groups affect both their
uniformity of size and their internal distributions. The last
consideration is fully treated in the statistical authorities; in
general, the best location of the limits from this point of view
is one which places the largest number of the observations
^ Cf. Yule, G. Udney, An Introduction to the Theory of Statistics^ pp. 79-83; King,
Willford I., Elements of Statistical Method, pp. lOS-106; also Bowley, A. L., Elements
vf Statistics, and Secrist, Horace, An Introduction to Statistical Methods,
CHARTS AND GRAPHS
COUKS FaCPESSORS' SAURISS
tal«rt«4 of full Profenaora In Colleses <103 Gni.T#fiUUi
(In public Inaillutlono)
Onlted State »
1920
j^Soyroat* t>. S» Bureau of Bducatlon)
(total nuabor of profotaoro, t.dSO. Arorago salary — arithm moan -• I S,126)
Fig* 275a Comparison of Fourteen Series Derived from the Same Data
by the Use of Different Group Limits and Group Sizes.
which belong to the group in or near the center of the group .2
Thus if we are counting men of various heights, and notice a
^This makes each group include all observations of doubtful accuracy, such as
the observations at and immediately about the round numbers.
FREQUENCY SERIES
317
COLLEGE PRoFESbCfiS SALi^IES
Salaries of Full Profet-rcirs in Collfges nr.d CTniTersitife*
(In Public Institutions)
United States
1920
(Source.- United States Turesu of Education; —
(iTote-- All date is for ISOO-groups - Senes C from f4Sl - 760, 751 - 1,050, «lC»J
Series h from $251 - 550, 561 - 850, etc.; - Senes J from $351 - 650, 651 • S50, etc*:
the total of each series being the same, 2,460.)
Runber
’ Salary
u> « «i
8|ggg§8888gSSg8§§S|8
*ft OK><00>cvtii><0>-«'4*r-Oe>«0 Cheu
Series
E
Number
Salary
10 10 10
O C4 O to
eidH U> 10 Id <0
4ltOQD<v>«O«C0ff
tendency of the records to bunch up heavily at the round num-
bers (which is only natural in such measurements) we should
do well to make our groups run from half inch to half inch,
so that each full inch will be in the center of its group.
Strictly speaking, the ordered list may be considered a fre-
quency series with such minute groups that there is but one
stub-value (however frequent it be) in each group; that is, that
CHARTS AND GRAPHS
318
each group contains but one value of the independent variable.
The series is generally unsatisfactory, however, because of its
unwieldy length and its many omitted groups or classes (that
is, classes with zero frequencies). We are therefore called upon
to make larger groups that they may be fewer in number.
As we increase the size and reduce the number of these groups,
we find the curve becoming more smooth in outline, the zero-
frequency groups disappearing. When carried out in detail,
the process is very like the moving-total operation which we
have seen performed on historical series, the same smoothing
out of insignificant wrinkles being the result. However, unlike
the historical series, there is no natural cyclic period to guide
us in determining the lengths of final intervals. Hence when
we have found the smoothest intervals, we shall take out only
the totals (not the moving totals) for publication. If the'
results are to be published to the layman it is well to adopt
round number intervals or class-limits, for his convenience,
however much the data may tend to ^Tunch up,’^ as previously
mentioned upon the round numbers. When the series is to be
presented ,to statisticians or used in research work, and such
‘Punching up” is noticeable, care must be taken to select inter-
vals or class-limits which will, so far as possible, place the
round numbers near the center of each class, and the class-
limits will therefore be fractions rather than round numbers.
Care must also be taken that the limits of the groups be
explicitly stated so that no confusion will result in the mind of
the reader. Thus it would be wrong to write *T00-200, 200-
500, 500-1000,” etc., in a table of the sizes of cities by popula-
tion. Such a series should be ^TOO-199, 200-499, 500-999,”
or ^TOl-200, 201-500, 501-999,” etc., as the case may be.
When fractions are present, as, for example, in a similar series
of the sizes of farms by acres, the best statement is ‘TOO and
less than 200, 200 and less than 500, 500 and less than 1000,”
etc.; but sometimes a shorter form, such as ‘TOO-199, 200-499,
500-999,” etc., will not be misunderstood. Whenever space
allows and there is any doubt as to either limits or the mid-
points of the range, two stub columns should be used, the
first to give approximate values of mid-points of each group
and the second to give intervals or group limits.
To the feature of uniformity of size of groups or classes,
much importance is commonly attached by statisticians, for
the convenience which will result in plotting and other analy-
FREQUENCY SERIES
3^9
sis. Obviously when a portion of the series contains groups of
half-inch size or range and other portions contain groups of
whole inch size or range, the two kinds of groups are not di-
FERCAFITA FIBS lASSES
io 74 Iftrgd ABerloea citios
1919
Tfire&pltHi Nuaher
Fire of
Lo»« cUiOt
nna«f $.60
10.5i-0.75
6
0.76-1*00
4
i.oi-i.ao
1
1.26-1*50
1£
1.51-1.76
10
1.76-2.00
6
2.01-2.25
7
2.26-2. 60
5
2.61-2. 76
&
2.76-3.00
a
3.01-3,26
3
8. 26-3,60
1
3.61-3.76
3
3.76-4.00
1
4*01-4.25
•
4.26*4.60
a
4,51-4.76
•
4.76-6.00
*
5.01-6.35
1
6.3626.60
*
6.61-6*76
*
8.76-6.00
*
6.01-6.35
•
6.36-6*50
•
6.61-6.76.
1
6.76-7.00
1
Over 7.00
1
The Frequency Series.
rectly commensurable and comparable. It is therefore always
a relief to discover that the compiler of a frequency series has
been able to adopt groups or classes with regular intervals
between their limits. The fact remains, however, that with a
very large proportion of business and sociological data the uni-
form group distribution is neither convenient nor satisfactory.
There are cases in which the entire range, as it is called, of the
distribution or series, is very great, and the great mass of
observations occur near one end.® To show the nature of the
3 “The general rule that intervals should be equal must not be held to bar the
analysis by smaller equal intervals of some portion of the range over which the fre-
quency curve varies very rapidly.” — Y ule, G. Udney, An Introduction to the Theory
of Statistics, p. 83.
320
CHARTS AND GRAPHS
distribution through this densely ^‘populated’^ portion of the
range, small groups or intervals must be adopted; but to pre-
vent an excessively long and tedious, often fruitless detail in
the remainder of the series, larger intervals must be used in
the sparse portions of the distribution. In such cases we are
obliged to alter the sizes of the intervals. Sometimes the
DUS AT ION OP STRIKES
United States
1921
(Source:- Ifonthly tabor Review)
Days of Duration Nmber
of
proximate
Range
Strikes
1/4
0 •
1/2
32
1
1/2 -
1-1/2
25
2
1-1/2 -
2-1/2
42
3
2-1/2 -
3-1/2
43
4
3-1/2 -
4-1/2
43
6
4-1/2 -
5-1/2
32
6
6-1/2 -
6-1/2
32
7
6-1/2 -
7-1/2
41
8
7-1/2 -
8-1/2
27
9
8-1/2 -
9-1/2
18
10
9-1/2 -
10-1/2
40
11
10-1/2 -
11-1/2
18
12
11-1/2 -
12-1/2
11
13
12-1/2 -
13-1/2
14
14
13-1/2 -
14-1/2
24
1&
-
18
14-1/2 -
18-1/2
69
19
-
21
18-1/2 -
21-1/2
42
22
24
21-1/2 -
24-1/2
16
25
.
28
24-1/2 -
28-1/2
30
29
•
31
28-1/2 -
31-1/2
31
32
*
35
31-1/2 -
35-1/2
34
36
-
42
36-1/2 -
42-1/2
60
43
•
49
42-1/2 -
40-1/2
37
SO
-
63
49-1/2 -
63-1/2
77
€4
-
77
63-1/2 -
77-1/2
57
78
-
91
77-1/2 -
91-1/2
55
92
-
199
91-1/2 -
199-1/2
166
Over
200
199-1/2 and over
42
Total
1,147
Fig. 278.
entire range is so excessively great that no two groups can be
of the same size, and it is necessary that the groups increase
progressively throughout the series.
In dealing with such unevenly-grouped series, the analogy
of the historical series is useful. What would you do if asked
to make a curve of a historical series, let us say, the world’s
production of gold since the voyage of Columbus, in which the
data covers at first centuries, then ages, then decades, then
quinquennial periods, and lastly, individual years. Clearly you
FRESIUENCY SERIES
jai
could not directly plot the points for the data and connect the
points to make a curve. You would have to change the data
to uniform time intervals before plotting the curve. You
SIZS OF FARMS
Oaited State*
1980
(Sottvoeto CeasusI
Aot-eage
fifiSS
than 3
£0,3SCh
S and less
than
10
268i422
10 "
0
a
80
607,763
20 “
o
II
50
1.603,734
60 «
#
u
100
1,474,768
100 •
0
u
175
1,449,668
175 ”
o
ti
860
530,795
260 ’*
U
w
500
475,693
SOO «
O
SI
1000
149,818
1000
ovar
«7,S87
tom e,<4MM
Fig. 279.
might sum up the parts of centuries into totals or even moving
totals for hundreds of years and so get a curve of 100-year pro-
duction. Or you could divide the earlier data so as to get the
OCLD ERQDUCTIOK OP THE
Estimated
1493-1919
{Soures:. Dnit«d States Statistical AlstraMj
Period
Value lA
Covered
Dollar#
1493 .
1600
501,640,000
1601 -
1700
606,316,000
1701 -
1800
>,262,806,000
1801 -
1820
194,216,000
1821 -
1840
229,320,000
1841 -
1660
1,696,909,000
1861 -
1870
1,263,015,000
1871 .
1880
1,067,569,600
1881 •
1890
1,074,950,500
1891 -
1900
2,101,240,900
1901 -
1906
1,613,098,600
1906 -
1910
2,167,604,800
19X1 -
1916
2,295,869,67$
1916
464,176,500
1917
419,422,100
1918
383,605,662
1919
366,166,077
Fig. 280.
average 10-year production in the earlier periods and sum up
the latter portions into ten-year groups, so getting a curve ol
production by decades. The intervals can be chosen at what
CHARTS AND GRAPHS
9
0
o e
O B
s
£
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ever size you wish, the point is that you must, for the sake of
the curve itself, convert the data into equivalent data for uni-
form intervals of time.
Fig. 281
FREQUENCY SERIES 323
tinoi) 39c*m‘9|
•4*40 » oooi> m*i9
(«<f*009} 2W6«
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Precisely
the same
’ Opel
• •HO UT »dnoat) •jo»-9x8uts u\
«aa «4 JO joi^nnN
frequency series with irregularly-sized groups. The fact that for
curves of cumulations (either of frequency or historical data)
this regularity of intervals is not necessary, sometimes makes
the cumulative curve, which we shall discuss in another chap-
282 ,
FREQUENCY SERIES 3^5
ter, far more convenient; but if we are to plot the uncumulated
frequency curve, and have unequal classes or groups we must
calculate the equivalents for equal intervals before we can
plot the curve. And if, as often happens, a terminal class
(group at one end of the series) be indeterminate, that is, have
no maximum limit, so that we do not know its group range
and cannot compute an equivalent figure, then we must simply
omit the last group from the series, leaving the reader of the
chart to conjecture that the curve runs off towards infinity.
Chapter XXVIII
FREQUENCY CURVES
Statisticians make a distinction between “discrete” and
“continuous” frequency series, which to the chart-maker is of
some assistance in determining the plotting points of the data
in a frequency curve.i Discrete data is that in which the inde-
pendent variable proceeds by leaps and bounds, lighting usu-
ally only upon the whole numbers or integers. When this
last is the case, the series may be said to comprise “integral
variates.” Thus buildings may be classified by the number of
stories or rooms they contain. Here you will find only regular
intervals of one Integer each, since fractional stories (barring
the so-called half story) and fractional rooms can hardly be
said to exist. Leaves may be classified by the number of their
ribs, flowers by the number of their petals, sales-forces, de-
partments, and establishments by the number of their em-
ployees, and cities by their populations. All of these cases are
examples of discrete series.
Continuous series are those in which the phenomena may
vary by infinitely small gradations, the data comprising what
are called “graduated variates.” Thus, of we examine the
height or weight of human beings, we find them varying by
the smallest possible amounts. Property classified as to value,
crops as to volume in bushels, tons, or the like, farms as to area
in acres, square miles, etc., and sales as to sizes,' are a few ex-
amples of continuous series. In business and economics much
the greater part of frequency data is of this type. And while
it is not always so, yet it is ordinarily in business statistics true
that continuous series require irregular group-ranges and
intervals, discrete series usually falling into equal groups. The
distinction between discrete and continuous data becomes less
sharp in those cases of discrete data which cover large
‘ Cf. King, Willford L, Elements of Statistical Method, p. 106.
326
FREQUENCY CURVES 327
ranges, such as cities classified by populations, for here it be-
comes necessary to adopt arbitrary groupings which are sim-
ilar to continuous data groupings. For small ranges the dis-
crete data usually requires no arbitrary grouping together, as
it automatically groups itself, and the continuous data is dis-
tinguished by the fact that group limits have to be arbitrarily
set for its distribution.
MEMBERSHIP OF STRIKES
Kumber of Persons Involved In Strilces
United States
1921
(Source;* Monthly Labor Revie v)
striking
Persons
Involved
Number
at
Strikes
Group*
Range
in Units of
10 Persons
Average
No. in
Equivalent
10* person
Class
X
y
Dx/10
y/(Dx/l0)
1-10
219
1
219
11 - 25
285
1.5
190,6
26 - 50
252
2,5
100.8
51 - 100
214
5
42.8
101 - 250
215
16
14.4
261 - 600
153
25
6.12
501 - 1,000
101
60
2.02
1,001 - 10,000
126
900
0.14
Over 10,000
14
...
....
Fig.
284.
Period Data.
For the cnart-maker, the more valid distinction of fre-
quency series is between what might be called ''point-data’’ and
"period data.” The former, point-data, is that which refers
to isolated, separate, and non-contiguous points along the range
of the independent variable. The mortality rates at various
ages are of this type and to this type belong a large class of
continuous series which comprise rates, ratios, percentages,
averages, and other comparisons between different basic fre-
quency series. Most discrete series may be classed as point-data.
Period data, to which most continuous series belong, is that
which covers connected and conterminuous groups or classes
along the range, applying throughout the groups from one
group-limit or interval to the next. The student is already
familiar with this distinction in the matter of historical series,
328
CHARTS AND GRAPHS
in which flow or stream figures, for example, cover periods of
time and stock or fund figures, for example, refer to points of
SIZE 0? FACTORIES
Manufacturing Establishments
Classified as to Number* of Employees
United States
1914
(Source:- United States Census)
Number of
Employees
per
Establishments
Employees
Establishment
Humber
percent
Number
Percent
a
32,856
11.9
....
1-5
140,971
61.1
317,216
4,5
6-20
54.379
19.7
606.594
8.6*
21 - 50
22,932
8.3
742,529
10.6
61 - 100
11,079
4.0
731,726
11.3
101 - 250
8.470
3.1
1.321,077
18.8
261 - 500
3.108
1.1
1,076,108
15.3
601 - 1000
1,348
.5
926,828
11.2
Orer 1,000
648
• 2
1,266,269
17.8
total
276,791
100.0
7,036,337
100.0
Fig. 285. Period Data.
time. Needless to say, point-data is normally plotted upon the
ordinates of the chart; period data is normally plotted in the
spaces between the ordinates.
VALUE OF MANUFACTURED PRODUCTS
Sstrabllahsanta Classified as to Value of Products, with Number of Employeea in Same
United States
1914
(Source;- United States Census)
Sstabli shments
Employees
'Value of Products
Number
Percent
Number
Percent
Dollars
Percent
Less than ^)5,000
97,061
36.2
129,623
1.8
233,381,081
1.0
45,000 and lesa than 420,000
87,931
31.9
429,037
6.1
905,693,168
3.7
420,000 and less than 4100,000
56,814
20.6
999,800
14.2
2,660,229,411
10.6
4100,000 and less than 41,000,000 30,166
10.9
3.002,071
42.7
8,763,070,135
36.1
41 # 000,000 and over
3,819
1.4
2,476,006
36.2
11,794,060,929
48.6
Total
276,791
100.0
7,036,337
100.0
24,246,434,724
100.0
Fig. 286.
• Period Data.
We come now to the last, and what is ostensibly the most
important division of frequency curves, namely whether the
FREQUENCY CURVES
329
curve shall be in staircase form or smoothed. The staircase
form, which really represents a collection of vertical bars, is
often called a histogram, and the smoothed-curve a frequency
HOURS OF LABOR
Number of Wage Earners Employed in Manufacturas
According to Prevailing Hours of Labor
United States
1914
(Source:- Statistical Abstract)
Hours of
Labor
Employees
Number Percent
48 and under
831,779
11.8
Between 48 and 54
944,562
13.4
54
1,013,079
25.8
Between 64 and 60
1,547,374
22.0
60
1,484,662
21.1
Between 60 and 72
249,026
3.5
72
106,080
1.6
Over 72
59, ns
.8
g. 287. Point-and-period Data.
polygon. Of course, as we noticed in historical curves, the
staircase form, if the number of steps or bars be great enough,
will closely approximate the smoothed curve in appearance,
ECONOifICAL SPEEDS OF TRUCKS
Maximum Speeds at which Loaded Trucks can be Driven
Without Reducing Life of Tires
Fig. 288.
Trucic
Miles par
Tonnage
Hour
^ and 1
19
4
4
17
2
15
13
4
11
5-7
9
Point-and-period Data.
and there remains little reason to maintain it. But for series
of but a few items or number of groups, the distinction between
the two is great and each has its special advantages and proper
uses.
The staircase curve or histogram is always more accurate
for period data, in that it preserves the exact areas underneath
330
CHARTS AND GRAPHS
the curve between each set of ordinates or group-limits. The
reader who recalls the belittling of area-representations for
charts in which we have early indulged in this book, may find
tmsavti. KRK Lsssss
In 74 Large ioarican Cities
19X9
Huttbor
of
O
M O
Dollare of For capita Los*
Fig. 289.
Showing how the smoothed curve varies from the staircased curve. The added
triangles (dotted) are equal to the deducted ones (black) in the aggregate, but
are not equal between any two adjacent ordinates.
this feature to be of little consequence. But statistical prac-
tise has it that the area is important in frequency curves. Of
course, if we plan to apply a planimeter or other area-meas-
uring instrument to the chart, the area is of real importance.
Otherwise it is generally to be relegated to the limbo of aca-
demic and scientific interests. We should, however, bear it in
mind, that we may the more correctly Interpret our charts and
base analysis upon them.
Not only is all period data more accurately represented by
the individual group-areas under the staircase-curve than by
the individual group-areas under a smoothed curve, but also
much point-data, if we class discrete series as point-data. To
be strictly accurate, discrete data should not be shown by a
connected curve at all, but by separate bars; for there are no
intermediate observations and the connection-line which forms
the curve has no meaning over intermediate spaces on the chart.
FREQUENCY CURVES
o n T
Nttmbdr of 1Ioa«n
> lO CHt O M ri
Number of Women
Children per Woman
'<ca«0'^iocoe»a
Children per Homan
SIZE OF FAMILIES
Number of Children of 1,000 Women
(married at least 15 years and having at least one child each)
British Peerage Statistics
(Source;- Yule, Theory of Statistics)
Fig. 290. The Staircased Form is Appropriate.
The chart-maker has largely to use his own judgment for plot-
ting discrete data, as he can almost equally well, for different
purposes, use the different methods of separate vertical bars,
staircase or bar-like curves, and smoothed curves, not to men-
tion plotting upon or between the ordinates.
By C. B. Davenport, Permission of Popular Science Monthly. ,
Fig. 291. A Very-Simplest Staircase Curve.
Showing the distribution of scallop-shells by number of ridges.
The disadvantages of the staircase form are many. In the
first place, as in historical curves, it is more difficult to distin-
guish a number of curves brought together for comparison
when they cross each other frequently. In the second place,
for period data, though not for discrete data, it is less significant
than the smoothed form. For while the data changes abruptly
33 ^^
CHARTS AND GRAPHS
from group to group, the phenomenon observed usually changes
gradually, the values usually merging between groups. This
is the more obvious if by a rearrangement of the original data
we produce more and smaller groups, for then the new groups
created take intermediate values. So to chart this data by a
staircase curve is to give a wholly meaningless sudden change
between groups, while to chart it by a smoothed curve is to
bring out to the readers of the chart more clearly the gradual
nature of these changes. In short the smoothed curve or fre-
quency polygon has a truer significance than the staircase form
or histogram, for period data.
EFFECT OF TUBERCULOSIS UPON LENOTH OF LIES
Kxp&ctancy of Lifo in Years for llhite Males vith and without Tuberculosis
and Consequent Shortening of Life Dus to ths presence of Tuberculosis
Metropolitan Life Insurance Company Industrial Policy-holders
1911- IS
(Source:- L. J. Dublin, Costs of Tuberculosis)
Loss due to
tutsrculosls"
Age in Years
Fig. 292. The Smoothed Form is Necessary.
For continuous point-data, that is, for point-data other
than discrete series, the smoothed curve is often the only pos-
sible form, the staircase form being out of the question. For
FREQUENCY CURVES
333
as the data represents observations at isolated points only,
along the range (the ^c-axis scale) it is to be assumed that for
intervening points intermediate values obtain, and the stair-
case form would be not only lacking in significance, but also in
accuracy. The dependent variable in continuous point-data
is usually the resultant of a process of comparison of two or
more different frequency series, being ordinarily expressed as
a rate, percentage, average or other ratio — a series of fractions,
if you will, in which the denominators are not constant. Point
data is to be found in a wide variety of forms, but is almost
always in essence a derived series of this sort. The processes
which yield point-data cannot be described as simply as the
BAIIK SALARIES
Salaries of federal Reserve Bank Employees under I500(^
Kew York City
1919
(Source:- federal Reserve Bulletin^
Individual#
of ‘’"ssssssssasa
Families — — t *- . '
Families —
r4
(1901)
gQO
«k
*
•
s-
I
^ 4()Q
1
1
i
o
%>
0
1
a#
800
1
1
1
1
1
1
1
■
1
M
i
1
m
1
1
1
1
1
■
■
■i
■
1 8 8 8 8 S
v> <n cv) to
^
f 1
t c\
* «
: 8
' *'
1 1
i 8
*
J 8
> u
> sr
1 t
> «<
^ 1
Dolls re of Salary
Fig. 293. It is DiHicult to Compare Two Staircased Curves.
processes which yield period data; for they are also of almost
unlimited variety and we shall not attempt their discussion.
It is worthy of notice, however, that during such preliminary
334
CHARTS AND GRAPHS
steps in the comparison of two frequency series we commonly
find the gun-shot plotting method useful.
suiting from the comparison of two series, is permissible only
Dollars of Salary
Fig. 294. A Cumulable Series.
FREQUENCY CURVES
335
in a broad sense, the point-data series not being a distribution
displaying the frequencies of the phenomena in the specified
groupings. Such point data, like the balances, stocks-on-hand,
or fund figures, in historical series cannot be cumulated, and
if
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the connection of the plotted points by a curve is a symbol of
the changes of the same phenomena through different condi-
tions, not a short-hand method of indicating dififerent and dis-
tinct quantities.
Fig. 295. A Non-cumulable Series
336
CHARTS AND GRAPHS
The question of staircase and smoothed curve plotting
methods have been given a somewhat lengthy treatment be-
cause it has afforded an opportunity to consider the dis-
tinctions between discrete and continuous series and period
and point data. As a matter of fact if there be but a sufficient
number of intervals or groupings in our series, the distinction
between staircase and smoothed curve plotting disappears,
and both forms of charts become alike and merge into a third,
PSRCAPITA FIRS LOSSES
In 74 Large American Citle*
1919
the rounded curve. The rounded curve is the true frequency
curve and is superior in significance even to the smoothed curve
or frequency polygon as the latter is to the staircase (rectilinear
or bar-form) curve or histogram. For the rounded curve not
only gives gradual change of values between plotted points,
but it also gives gradual changes of the rates of change of these
values* The derivative of. a frequency polygon would be a
histogram, but of a rounded curve would be another rounded
curve or at least a smoothed one.
We have not, however, laid much emphasis upon the
rounded curve, because if the data be sufficiently detailed it
will be approximated by either staircase or smoothed curve
plotting. Some authorities recommend the artificial rounding
FREQUENCY CURVES
337
lOHKMBH’S COMPEtrSATIOS
Delays in Making Payment for Claims in Three States
Hew Yorlc (state and private insurance)
Ponnftylvania (state and private, except self insurance)
Massachusetts (private insurance only)
(Source*- Monthly Labor Review)
(Figures show percentage of total cases, 137 in N.Y,, 4,093 in Pa. and 186 in Mass.)
Fig. 297.
Computed averages must be used for the irregular intervals.
of curves.2 This is to be done either free hand, or by a curving
ruler (called by draughtsmen a “French curve”), taking care
in either case to pass the curve through all known values (i.e.
plotted points), but making the remainder of the curve as
little angular as possible. The result is almost always more
interesting to the casual reader, obviously because of the more
faithful portrayal of the nature of changes from interval to
2 ‘The object of smoothing is to eliminate accidental variations and establish
normal tendencies.'’ — King, EUments of Statistical Method, p, 108.
CILiliTS AND GRAPHS
3 J 3
AdSS OF HOOEASDS AND WIV®
ProbabXo A£« of Wife oooordlni; to AX* ot Ru*b&nd
Greet Britain
1901
(Bource:- lule. Theory of Stetlatlce)
Ojuper Cueriil*
OOa>coM*o«o«<<n<«>t^r>'40
Uedlen OrtOi4r-dioOy‘a>.o»“<-<io
<VtNNi4«Q-^'«t0l0in<0«>r~t-
iOl<>lO»««^ 0 »iOt'jOO«M<vloe«l^
Lower Quart! le l;SjSS‘^;g:j!§gJgS^25SS
90
«0
70
5 60
^40
30
20
10
lOOutOiOpiaOioOiOOioOio <>
Age of Husband in 'Yeara
Fig. 298. A Zoned Frequency Curve.
interval. But it is a dangerous practise for the beginner to
round his curves artificially, being a wholly inspired embellish-
ment and often leading to slipshod execution. In the early
work of the student, the smoothed curve is all that should be
attempted, for it is all that the data establishes. And in the
research office, it is for the same reason about all that is neces-
sary or safe. In fact, in the research office, it is often sufficient
to plot the points only and omit altogether their connections
which form the curve, since it is often desirable to superimpose
thereon rounded curves of a theoretical nature.^
In general, the problems in the graphic presentation of
curves arise, first, in the selection of the independent variable
^Needless to say, composite curves may be drawn for frequency series as for
historical series, in the various forms which have been described (see Chapter XIX),
Thus, we may have relative and absolute band*charts, gun-shot plotting, and even
vertical and horizontal bar-charts.^
FREQUENCY CURVES
339
FElklALE OCCIDENT liORTi^LlTY RATES
Oe&th-ntes oer 100,000 of Population of Pemalea of Each Ago
For Specified Accidonts
United Statea
1910-1912
(Source - Mortality Statistics, United States Census)
nil Accidents
Miscollfineoua
BaiLwiys
Cromimg
Pail*
70
60
EO
40
30
SO
10
0
<
Fig. 299 . A Frequency Band-chart.
and the processes of compiling the frequency series ; second in
the establishment of group limits for the groups in the series
and the conversion of irregular intervals into corresponding
equivalents ; third, in the plotting of data on the ordinates or
between them and lastly, in the use of the staircase curve (his-
togram) or smoothed curve (frequency polygon). To the solu-
tion of these problems the distinctions between discrete and
continuous data (the integral and graduated variates) and be-
tween period and point data bring some assistance, but no set
rules of thumb can be given, to which exceptions may not be
found. In the wide field of frequency curves, to which the
historical curve stands in the relation of a small but Important
part, the chart-maker and the statistician must rely largely
upon native judgment and the precepts of his individual ex-
perience.
34 °
CHARTS AND GRAPHS
OUTPUT OP WORKBPS
Relative fiRurea of averer.e dally oulpat of manual workers
in stereotype!' end non-sterectypr d oferptions
(Total number of workers stereoptji d, ? 1 , not sterect. 29 )
(Source - P. Sargei.t Flcrorco)
(Output cf med*an worker * ICO )
Stsreetvped
Rot sterecljped
100
,
_J
r
1
Jl
J
r
t
i
~
~n
j
1
1
i
X
-J
J
._i
L
n
u
]
/
i:
/
iQ
SooS SoJotSSSoS
Volume cf output (relative figures)
Fig. 300.
A ‘‘Relative’^ Frequency Curve is as possible as a relative historical
one
Chapter XXIX
OGIVES
If moving totals are your strongest weapons in the analysis
of historical statements, cumulation is your trump card in the
analysis of a frequency series. Where the series is continuous
(as explained in the previous chapter) the process of cumulation
does away with the need for the staircase curve, and gives us
a smoothed curve which is far more convenient. And for both
discrete and continuous series, the curve of the cumulated data
is one which can be easily compared with similar curves, re-
gardless of diiEFerences of scale figures or group units. The
curve of cumulated frequency is called an ‘^ogive,’^ from its
resemblance to the outline of a shoulder. It runs diagonally
across the chart, generally in an ^^S”-shape.
SIZS OF FAflMS
United States
1920
(Source:- Census)
Simple series "Less- than” cumulation
Acreage
Numher
Acreage
Number
Loss
than 3
20,360
Less
than 3
20,360
3, less
than 10
268,422
10
288,772
10,
It
20
607,762
-
20
796,534
20,
«
fl
SO
1,503,734
«
60
2,500,268
50,
n
f9
100
1,474,753
ID
100
3,775,021
100,
fi
«
175
1,449,659
»
i>
175
5,224,680
175,
n
a
260
5^0,795
It
It
260
6,755,475
260,
»
t»
500
475/692
9
n
600
6,231,167
600,
«»
9
1000
149,812
It
a
1000
6,380,979
1000
and
over
67,387
Total
6,448,36$
Fig. 301. Tlie ‘‘Less-Than’’ Cumulation*
341
34 ^
CHARTS AND GRAPHS
While k would be meaningless to cumulate a historical
series backward, and we therefore cumulate historical series
only in one direction, it is possible to cumulate a frequency
series from either end of the series. If the cumulation begins
at the lower end of the data, it is called a ‘%ss-than^’ cumula-
tive, for the sub-total or progressive cumulative figure repre-
sents the number of items having less than the maximum quali-
fication of the last added group. If the cumulation begins at
the upper end of the series it is called a “more-than^^ cumula-
tive, the sub-total or progressive cumulative figure representing
SIZE OF FAHidS
United States
1920
(Source:- Census)
Stmplei
series
cumulation
Acreege
Number
Acreage
Number
1000
and
over
67,387
1000 and i
over
67,387
500 and leas •
than
1000
149,812
£00
217,199
2eo
**
soo
475.692
260
n
692,891
175
260
630,795
175 ”
1,223,686
100
175
1,449,659
100
2,673,345
60
100
1,474.753
50
4,148,098
20
n
50
1,603,734
20 ”
i»
5,651,832
10
,r
20
607,762
10
n
6,159.594
3
"
10
268,422
3
6,428,016
La-.s
t>-an
3
20,350
Total
6,448,366
Fig. 302. The “More-Than” Cumulation.
the number of item having more than the minimum qualifica-
tion of the last added group. It is often useful to prepare both
the *^more-than'^ and "less-than’’ cumulatives for a frequency
series and to plot them both on the chart as well as the series
itself.
It is one of the great advantages of the ogive that by its
means a frequency series may be graphically presented and
analysed whether or not the groups of the series be uniform in
size (group-range). There is no labor of calculating values for
equivalent groups. All question of staircase curves likewise
disappears, even discrete series, when cumulated, being prop-
erly shown smoothed. It is another advantage that several
343
ogives can be easily shown together and compared upon the
same chart. The various series need not have uniform and
identical group intervals. The ogive is therefore the only
feasible method of comparing frequency series which do not
have the same groupings. A further benefit is that the many
ogives will generally be found to intersect very little, so that
the confusion which attends superimposed frequency curves
is avoided by ogives.
For anatytical purposes it may not be amiss to note that
the median of a series is shown by the intersection of the two
EXPECT AH CY OF LIFE
For Adults without Tuberculosis
Registration Area, United Statsa.
1910
(Source:- L. I. Dublin; Cost of Tuberoulosif )
Years
of
Ago
20
25
30
35
40
4'5
60
65
60
65
fo
n
6d
65
90
95
Average
After Lifetime
46.$
42.3
38.1
33.9
29.9
26.0
22.1
10.5
16.2
12.1
9.6
7.2
5.4.
4.0
2.9
1.9
Fig. 303. An Example of a Frequency Series (So-called)
Which Cannot be Cumulated.
Ogives, the more-than and the less-than, for the series, or by
the value of the abscissae at the intersection of the curve with
the ordinate of half the height of the 100% ordinate; while the
mode is shown by the portion of the ogive in which the slope
is steepest. These are statistical rather than charting concep-
tions. The median may be described as the middle or central
344
CHARTS AND GRAPHS
observation, and the mode as the most common observation.^
We may also note that two minor variations of the two
cumulatives obtain, which depend in part upon the plotting
SIZE OP FARMS
United States
1920
(Source ~ United States Census)
(Figures shov number of Farms containing more than and less than speoifiod axsaber of acres)
and in part upon the nature of the data. These variations are,
for the “less-than” cumulative, a “less than and including”
^ Readings from the curve, for the median, decils, quartiles, or percentiles, when
secured by interpolation from the curve and not from plotted points on the curve
give values, but of course do not give cases. The median case, for example, can be
found only by reference to the original data, and exists only if the total number of
frequencies be odd. The median value, however, is the intersection of the curve with
the 50 per cent abscissa, and is obtained with increasing accuracy as the frequency
groups are taken smaller and smaller, and the curve itself plotted in greater detail.
Oneorreotod CvimulatlToa Corrected
a 3 = 3 Cimulattvse
OGIVES
345
SXZS 0? FAHHIIS
Rvffiber of Children df 1,000 Women
(married at least lb years and haring at least one child each)
Sritish Peerage Statistics
(Source.- Yule, Theory of Statlatlos)
Figf. 305. Showing the Four Possible Cumulations For Point Data.
To find the median or quartiles, etc., graphically, it is necessary to use the dotted
line showing the averages of these cumulatives the “corrected cumulatives").
0 .&
346
CHARTS AND GRAPHS
cumulative (stated concisely as “-and less”) and, for the
“more-than” cumulative, a “more than and including” cumu-
lative (stated concisely as “-and more”). They are essen-
BOORS OF labor
Muober of Wage-aarnora Employad In Manufacturet
Aocerding to Provailing Hour* of Labor ,
United States
1914
(Source'- Statistical Abstract)
61.88
16.55
iM
30.15
83.65
98.45
T4.a
26.9
2.9
ee .3
48.9
5.8
0.9
11.8
61,1
94.2
99.2
Uuaber of Boor*
Fig. 306. The Four Possible Cumulations for (Point-and-) Period Data.
To find the median or qiiartiles, etc., graphically, it is necessary to use the dottei
lines of the averages between these cumulations (i>,, the “corrected cumulatives”).
tially due to differences in the plotting points of data, and are
sometimes more suitable for cumulations of discrete data.
It has not generally been observed that the ogive is really
simpler in its nature than the frequency curve. Because we
have secured the frequency series by a very careful grouping
OGIVES
347
HOURS OF XlBOR
ffUfflhflr of W9R<»-earner» Employed In llandfadtUfM
According to Prorailing Jlour# of Ld1)or
United States
X914
(Source:- Stotlatical Abstract)
Percent of Waga-oarnore
Worlcing more than and
Including each Speolflod
Rumba r of Hours
Percent of Wage-earner*
Worlcing less than and
Including each Specified
Number of Hour*
Percent of Wage- earner*
Worlcing each Specified
Number of Hour*
Vumbar of Hour*
100
90
60
fO
eo
60
40
60
20
10
0
U <0 -r o <a oi u
« iC ¥> <0 <0 «- «»
I -
Fig. 307. The Rounded Ogives.
Showing that median and quartiles, etc., cannot be easily found from two un-
corrected ogives.
of our original data, arranged in order of magnitude, and have
then derived the ogive data from the frequency series by cumu-
lation, we are prone to think of the ogive data as a somewhat
more advanced and possibly more puzzling form of statistical
series. The fact is, however, that the cumulation merely
brings about a reversion to the original data in order of mag-
nitudes, somewhat condensed as a result of the groupings. If
we lay off the original data in the form of a bar-chart we will
see at once that the ogive is merely a smoothed curve passing
through the ends of the bars. It is for this reason that the size
of groups or their uniformity is of no importance in making
the ogive-chart, smaller groups merely defining the ogive-curve
more precisely .2
- Cf. Robert E. Chaddock, in the American Statistical Association Quarterly, June
1921 , p. 769 ff.
34^
CHARTS AND GRAPHS
tolluTB ot I>«r(japita loss
35 °
CHARTS AND GRAPHS
SIZE OF FAMILIIS
number of Children of 1,000 Women
(married at least 16 years and having at least one child each)
British Peerage Statiatice *
(Source:- Yule, Theory of Statistics)
Ifothera haring
each specified
iTinflber of Children
Uothers having up
to and including
each specified
I^u^l■ber of Children
t-
ta
fON iocMOW
irt cvi cy 1-4
*00
U>»-4 r^.^^OOO•-^•-4
«-4 CM »0 <0 t- CO
^ to to «n
to fo no o» <30
o> cn m O)
CR
Oi
Mothers having more
than and including
each specified
Kuaber of Children
o
o
CO
o
a»
CO U> -f >~f
lo <>-1 d> CO
to to lO CM
to «d*
to «-«
<v»
Where individual series are to be analyzed by themselves
the horizontal scale for the ogive can be the same as the hori-
zontal scale for the ordinary frequency curve from which the
ogive has been derived and for which it has been substituted.
The vertical scale, however, will have to be condensed so as to
include the total of the entire series. But because a frequency
chart is usually designed to show the comparative behavior of
the phenomena studied, it is often useful to turn the actual
data into percentages and to use on the chart a vertical scale
calibrated in percentages. The percentage values are more
useful for generalization and ready comparison with other
ocn-'ES
3^1
DUStATlCl. CF EMPLOYVEJT
Percent Lietribtticn of 1,30S yal« ard H4 Par-ala Ehsployeas cn ih* Payroll
an4 2.618 Male and 63 Fecale Separancne wl-o had Served aio-e than Specified Periods c? Tta«
Califorrie Sugar Sefirery-
Acure . May 31, 1S18
Separated - April 1, 1917 - May 31, 1918
(Source - Paul F. Brtsserdcn)
ogives, than the absolute scale figures of one particular series
of observation or samples. For similar reasons you may find
it desirable to turn these group-divisions which form the hori-
zontal scale also into percentages, both in the data and on the
chart. In both cases the percentages are percentages of the
total or maximum limits of the series. When several frequency
series are being compared, and the series differ both in the
total number of observations or items in the series and in the
group-divisions or group units into which the series is divided,
this little trick of turning all readings into percentages may
be very useful, as by its means you can chart the ogives or
cumulatives of all the series upon uniform chart-fields. The
fields on which the curves are to be plotted should generally be
CHARTS AND GRAPHS
and Hour® of T<Mian
Weekly Wage-rates and Hours of Labor of 3,720 Women
In Department Stores and Dry-gcods and Uillinery Eetabiishaents
Virginia
April 1, 1920
(Source*- Uonthly Labor Review)
(figures show number of women receiving more than specified wages
apd working for more than speoified number of hours.)
.HI (O OS to <4* Q
r, , , „ jOi!0«-tO>carH
Dally Hours Jo <3 ^
Figf. 312. Comparison of Absolute Data is Sometimes Difficult*
square, running from zero to one hundred per cent along both
axes of the chart. Needless to say, the fields should be uni-
formly positioned upon the sheets of paper so that the various
charts to be compared may be freely subjected to ‘‘light analy-
sis,'' that is, to the method of analysis which consists of holding
two or more charts together up to the light to detect the varia-
tions of their curves.
In the ogive chart we first meet with a type of chart which
illustrates at the same time two different sets of figures for
the same curve. There should therefore be space for data
OGIVES
353
not only above the chart but to the right of the chart, and the
chart field should not be placed close to or near to the right-
hand margin of the paper as was the case in historical curves.
The data above the chart is obviously the original data of the
cumulative from which it is plotted, each data figure being
placed above the ordinate or corresponding scale figure on the
WAGES OF OFFICE, SALES, AJJD SHOP WOBSSRS (JiAlE)
W.tJcly lAges of 932,808 Wago-oornors, 63,619 Bookkeapara, Stanographar*,
and Offica Clarks, and 23,766 Salaaman (not travailing)
(All Ualaa 18 yaara of age and over)
Maniifacturing Indvatrlaa
Ohio
1919
(Flguraa ahaw parcantaga of total racairing leae than each tpeoifiad mga}
(Source:- Xnduatrial Comi&iaaion of Ohio)
Salat
Clarka
Earnara to S ® 81
Fig. 313. Comparison of Relative Data is Easy.
x-axis of the chart. The data to the right of the chart field
will be secondary or derived data, obtained by taking the
readings or values of the curve at each of its intersections with
the abscissae or horizontal rulings, using the corresponding
scale figures of the y-axis for the new stubs, and taking the
corresponding values along the x-axis as the new or derived
data. This secondary data forms a new table of the same
phenomenon rearranged so that the second variable or de-
354
CHARTS AND GRAPHS
pendent has become in a sense the independent one and its
values would appear as the stubs in a retabulation.
The ogive-chart is excellently adapted for the process of
interpolation. The derived data just described are an example
of this use of the chart. Interpolation, of course, is the name
given to the process of reading new values between originally
given values. Thus, by means of the ogive-chart, originally
incomplete data can be filled out with interpolated figures.
But the interpolated figures of course do not have the same
kaoss or orriCE lOBiisa (rnuu)
IttJtly «**•• e* Muiv »o«Wt*«p*r». Sv*negr.f>h«r*, »ti4 Oftle* CUtK*
(U T**r» *r *(14 Ov»r)
Itanufaeiurinc Induitri**
or.l»
(Vtiur** »h*w paraant. of foUl roool*tm U*f Uuin »««h tp»clfU4
(touroo,- Induttrur ComUstioB of Ohko)
5
a
I ; *« e « U « u *
•k of
^ (Ovstrt) ton poroont
of oawn
^ m e>tr t.'&.SO
Sf.tO
22.00
20.10
X7.7»
K.eo
UM
Drir 42 s. SO
22.00 • 20.60
20.60 • 22.00
10.00 • 20.(9
17.7( • 1».00
1(.(0 . 17.71
U.t( • U.KO
U .»0 O IC .»
12 .tQ
U . S 0 . u.to
ii.eo
OniMT tU.«ft
degree of accuracy as the original data, being made on the
theory that the line drawn between the plotted points of the
original data has been correctly drawn. In spite of their pos-
sible inaccuracies, they are often extremely useful in reducing
otherwise incomparable frequency series to comparable group
units or to uniform percentages. Where the chart itself is to
be used, the interpolation is not necessary, the connecting
lines which form the curve being in themselves plottings of
interpolated values. But when the chart will not be presented
in the final report or summary of the case, the interpolated
figures are necessary, the interpolated figures being obtained
from the chart before the chart itself is discarded.
OGIFES
355
The ogive-chart is one of the most important and generally
useful of the non-historical chart-forms and we will have oc-
casion to return to it in the future with various elaborations
and improvements.®
® It is thoroughly regrettable that statistical practice has so consistently considered
the range of a frequency series to be the independent variable and its frequencies the
dependent. For while this is logical enough in the simple curve, it introduces into the
ogive or cumulated curve, absurdities which make the latter not only unnecessarily
obscure to the layman, but also brings about an unjustifiable violation of the primary
rule in curve charting that the plot of all points should be independent Xy dependent y.
In the simple frequency curve, we are interested only in frequencies and they are
obviously and properly dependent. The reduction ad initium of this chart to a bar-
chart would, as has been seen, require the use of vertical bars, and the curve is but
the short-hand connection of the tops of these bars.
A very different case is that of the cumulated series. Here the frequencies are
only in an immediate sense dependent; in the last analysis, they are independent and
the range is dependent. As has been seen, the return of this curve to a bar-chart
would require the use of horizontal bars — a fact which clearly illustrates the really
dependent nature of the range.
Statisticians have, however, so short-sightedly adopted the ogive as a derived
chart (by cumulation) from the simple frequency curve, that they have followed its
arrangement of the variables on the chart; and as the ogive has been confined to
use by statisticians, the practice has become so settled that to advocate a change at
this time would seem only to be adding confusion to a science which, more than all
else, is in need of standardization.
Life does not, however, always confine itself within academic rules, the round-
about path must always be supplemented with a fence; and we venture the prediction
that in time, as the ogive comes into commercial use, this arrangement will be scrapped
and the ogive plotted on ;t-frequencies, y-range, passing first through a period of
confusion which we do not seek to bring about. But he who must tell his story clearly
or not at all will drop the old arrangement, A step in this direction, though perhaps
un'concious, is that of the publishers of probabilities paper for ogives, who probably
only by accident or for convenience have calibrated their scales so as to give
A:-frequencies, y-range. The chart in this form is more intelligible to the layman.
Chapter XXX
LORENZ CURVES
Neither the frequency curve nor its ogive have that pecu-
liar tang of popularity, that engaging frankness which appeals
to the “average man.” If we consider the ogive the simpler of
the two, since it is merely a curve passing through the ends
of horizontal bars, then indeed it is a curiously unscientific
chart, in which the usual position of the dependent and inde-
pendent variables is reversed. The frequency curve is then a
short-hand method of arraying these, with a large degree of
chance in its formation when various groupings have diflPerent
results. And if we consider the ogive as a cumulated frequency
curve it then involves all the obscurities of the simple frequency
curve, augmented by further complexities of its own. We
otJim or FACiosiss
VtimfcHir ojT l&zxafaoturixig SstabliafaDunts
of apeolflod Bizaa
(tlze 1)01212 2Maa\irod hy valu® of prgduotf)
TIzilted Siaiat
19U
8ouro0t . S « Qeziaut
Spaolfiod
Valu0 of Products
per
Bstahlislsaent
•
Humber
of &11 Hstablishments:
haTing apooified value of products
Humber
per 0 eat
Less than 15,000
97,061
15«2
16,000 - 820,000
57.921
11.9
120.000 - #100,000
56,814
20.$
#100,000-4l»000,000
10,166
10.9
#1,000,000 Olid over
1,819
1.4
-TOTAL
275,791
100.0
Fig. 315. The First Measure — By Count of Items.
356
LORENZ CURVES
357
now take up, therefore, a curve which has more popular ele-
ments in it, with a consequent sacrifice of statistical detail.
OUTPDT OF FACTORISS
ViLlxid of.Produots of Manufacturing Batablisfaaonta
of apooified alzos
(also being measured by value of prodaota)
United States
1SX4
Souroe: — O.S*Census
specified
Value of Products
per
Establishment
Value of Products
of all Establishments
having specified value of products
Dollars
Percent
Loss than $5,000
223,30L,O81
1.0
$5,000 « $20,000
905,693,168
3.7
$20,000 - $100,000
■ J?,660,229i4n
10.5
$100,000-51,000,000
8,763,070, 1«5
36.1
$1,000,000 and over
11,794,060,929
48.6
TOTAL
24,246,434,724
100,0
Fig, 316, The Second Measure — By Count of Units.
All frequency distributions afford two possible series for
precisely the same data. The first and more usual series is the
count of items in each group of the distribution, the second
OUTPUT OF FACTORIES
Humber and Value-of-Produots of Manufacturing Establishments
of specified sizes
(size being measured by value of products}
United States
19U
Source:- U.S. Census
Specified
Value-of -products
per
Establishment
All establishments having specified value-of-produets
Ntimbor
Value-of-products
Dollars
Percent
Less than $5,000
97,061
235,381,081
1.0
$6,000 - $20,000
87,931
31.9
905,693,168
3.7
$20,000 - $100,000
56,814
20.6
2,550,229,411
10.5
$100,000-$1,000,000
50,166
10.9
8,765,070,135
86.1
$1,000,000 and over
3,819
1.4
11,794,060,929
48.6
TOTAL
275,791
100.0
100.0
Fig. 317, Both Measures.
358
CHARTS AND GRAPHS
and alternative series is the count of the units of measurement
attributed to these items. Thus a classification of farms by
their size (in acres of land) can show us either the number of
farms of each size or the aggregate number of acres in the
farms of each size. The data of cities classed by their popula-
tion may count either the number of cities of each specified
number of inhabitants, or it may count the inhabitants residing
in these cities. The Census of the United States, in its analysis
of the manufacturing establishments of the country, according
0OTK3T OF FACTORIES
tJRd Vuluo-of-Producta of Manufacturing Establishmenta
of specified alaa
(cite being measured by value-of-produots}
United States
XS14
(»otej— Ali daU in percentages of total or aggregate)
Source: «*• U. S, Census
Spaoificd
Valua - of -Pr oduo t s
per
Est&blishmai^t
All Establiehaenta having
specified value-of -products
Number
•Value-cf-Pro ducts
lesa than t5|000
S6.2
1.0
15,000 • $20,000
21.9 :
3.7
less than |20,000
07.1
4,7
$20,000 - 1100,000
20.9
10.5 {
Lt35 than 1100,000
87.7
15.2
1100,000-11,000,000
10.9
16.1
Less than 11,000,000
51.5
$1,000,000 ana over
mm
' 46.6
itoy value whatever
100.0
100.0
Fig. 318 . Cumulating the Percentages.
to their employees, gives both the number of establishments
and the aggregate number of employees in such establishments ;
in its analysis by value of products, gives both the number of
establishments and their aggregate value of products. Ex-
amples might be multiplied without end, for whenever we dis-
tribute the items of any phenomenon into groups upon the
basis of some units for measurement, we are then at liberty to
count either the items themselves or their units of measure-
ment, group by group.
LORENZ CURVES 359
The thought, therefore, occurs to us that a chart could be
made in which one of these series serves as the independent
variable for the other, and in which the two values for each
OUTPUT OF FACTORIES
V&lue-of -Products wad Numbcr-of-Eotablishm&nts
of Itonufacturing Establishments of specifiad siz®
(size being measured by value-of-produats)
United States
1914
(Kote:- All data in percentages of total)
Source:- U.S. Census
Specified Size
as shovm by
Establishments having
specified value-of-products
per
Establislmeni
Ntmiber-of-
Establishments
Value-pf-
Produots
Loss than $5,000
35.2
wmm
Loss than $20,000
$7.1
mm
Less than $100,000
87.7
15.2
Less than $1,000,000
98.$
51,S
Aiy value whatever
100.0
100,0
Fig. 319. Data For the Lorenz Curve.
group of items are made the co-ordinates of plotted points.
Obviously, if we are not to have the curve which connects
OUTPUT or PACTCSUES
The Value of Produete of
ipeelfled groups of )Canuf&oturi»g Sitehllahnentt
United Siatee
1914
Xu percentage figures
(Koter*- ill groups eomposed of estahllshmentf hariog,
least value of produotl.)
Source:— U 6 Oensut
Ktoiiber of
Value of
SstahlishRients
Produett
percent
Percent
35.2
1*0
67<1
67.7
96*5
100.0
100.0
Fig. 320* Data to Plot the Lorenz Curve.
these points moving backward and forward as well as up and
down, we must cumulate the series which is to be used as the
360
CHARTS AND GRAPHS
independent variable, and since it will be found that the curve
has greater significance when fcoth are cumulated, we in-
variably cumulate both series. It is then a matter of indifFer-
OUTPOT OF FACTORIES
The of Products of
specified groups of Manufaofcuring Eatsiblisfamentt
United States
19X4 '
In percentages
(Note:*— ^ All groups ooBjposod oumilativoly of the
eatabliahments havirg least value of products)
3ouroe:*«o U. S. Census
0 10 20 30 40 60 60 70 80 90 100
Percentage of aggregate number of Establishments
Fig. 321. The Lorenz Curve.
ence which series be used as the independent one, and either
series may be plotted upon either axis of the chart. It will be
seen that the chart is closely related to the ogive, since it uses
oirrm tr facwrib
Vnitsd States
19 li.
7h«
n
Isr^est 10 peroent of the
factories preduoe 78 percent of the values
" " 87 " " *1 "
the
smallest
90.
ao,
only 22.
" IS.
m
50 "
H
**
n
"
"
"
"
w
70,
"
7.
n
40 ■
"
«
**
9«
II
"
n
"
60,
**
4.
H
m
half
97
n
*•
n
tt
II
S.
ft
60 "
**
"
*•
98
••
n
n
*40,
'*
2.
tl
•*
70 ”
n
*•
e
99
n
H
m
n
50,
about 1.
n
tt
80 "
n
n
**
99f
*'
n
n
n
20,
4.
It
n
90 "
"
H
99|
*
"
«
*
10,
”
ir*
(Kota;— “Largo” and "small" refer to the si*e of the factory aa measured by the raluo of Its produotto)
Also
10 percent of the values are produced by the largest ^ percent of factories'; 90 by the aioAllest 99jr
20 " " " " " n » « II II « " j 80 ^ " " 99.
•ioii;. .. • »*•
Fig. 322. What the Lorenz Curve Tells the Layman,
LORENZ CURVES
361
a cumulated series; it will also be seen that the chart omits
altogether the classes or groups in which the data has been
collected. Lastly, to produce a uniformity of these charts, and
to facilitate the comparison of different distributions upon the
same chart, all items are turned into percentages of the totals,
and plotted upon percentage scales along both axes.
When this type of curve is drawn upon a square field, with
equal percentage scales upon each axis, it takes the shape of
an archer’s bow, and the curvature of the bow has a peculiar
LOBENZ CUBVE SHOWUSTO THE BISTBIBUTIOK OF
INCOMES IN 1918.
Front Income in the United States,” hy the National Bureau of Economic Research, by permission.
Fig. 323. The Familiar Example.
significance as an index of dispersion in the original distribu-
tion. For a little thought will show that a uniform distribu-
tion in which all items are alike will yield not a curve, but a
straight line. The first ten per cent of the ''population” in the
series of personal incomes, for example, if all incomes were
equal, would have ten per cent of the total income of the
country, the first twenty per cent would have twenty per cent
of the total income, and so on. Hence, the distance between
the curve and the straight-line diagonal indicates the degree
in which the series is removed from a perfectly uniform dis-
362
CHARTS AND GRAPHS
tribution — a feature which statisticians call dispersion or
scatteration.
The Lorenz curve, as this form of chart has come to be
known, has not been much used except in the analysis of in-
come and wealth distribution, but it is obvious that it is
capable of use for any and all frequency series. It is simple
OUTPUT OF FACTORIES
The Value of Products of
specified groups of Manufacturing Psiabllshnonto
United States
1914
In percentage
Source:— U. S. Census
Percentage of aggregate number of SstabXishmenta
Fig. 324# Two Curves of the Same Data By Using Both **Morc-than’^ and
'^Less-than'^ Cumulatives.
and popular in its appeal, without being in the least Inaccurate
or meaningless. It has certain advantages in the emphasis it
throws upon dispersion and unequal distributions. Its chief
disadvantage is in the omission of the group-by-group data
for the series it illustrates, but this data is more in the nature
statistical detail, and does not belong to what may be called
a summary analysis of a distribution; to the average man such
detail is confusing rather than helpful, while the results of the
LORENZ CURVES 363
dispersion, which this chart shows, form in his mind the meat
of the matter.i
The principles of the Lorenz curve can, however, be ex-
tended to innumerable comparisons between frequency series.
It is not necessary that the two series compared be the two
alternative forms for the same data. Though the latter is
usually the sounder practice, there may be occasion to bring
ODIPIII or FACTOSXSS
Th* Value of Produote
of «peoified groupe^of Mawafacturing Establlilmiist*
as 4 of speoifled groups of Employee* tberelA
United States
1914
In percentages
Souroe; U* S. Censu*
together series which, for example, have different units of
measurement. Thus the manufacturing establishments of the
country are classified in the census as to number of em-
ployees, value of products, value added by manufacture, horse-
power used, and the like. Taking any one of these classifica-
^ As no data can be easily appended to the Lorenz curve, unless we elect to give
readings of the curve at various points (in which case data belongs at both the top and
on the right side of the chart, to give readings for both variable scales), it is generally
sufficient to append to the Lorenz curve the percentage cumulations from which the
curve has been drawn. These afFord to the inquisitive full details for the group-by-
group distributions which the chart itself does not show.
364
CHARTS AND GRAPHS
tions, the census gives the other features just mentioned, for
the establishments forming each group in the classification.
The true Lorenz curve will then bring together, for example,
the number of establishments having a specified value of prod-
ucts and the group value of the products of these establish-
ments. If we like, however, we can bring together the value
of products of each group and the number of employees at-
tached thereto, or the value added thereby, or any other
OUTPUT OF FACTORIES
*Valu« of products" and "Value Added t>y Msmifacture"
of specified groups of Manufacturing EstaUllshmenta
and of specified groups of Emplcyeea therein
United States
1914
In percentages
Source: 0. S. Census
Fig. 326. The Logical Form is Triangular.
feature we desire to show. This forms a pseudo-Lorenz curve
which, though slightly more complicated in principle, has the
same popular features.
Popularity is the main feature of the Lorenz curve, but
it is not without its scientific significance. As already re-
marked, the deviation of the curve from a straight line shows
the dispersion or scatteration of a series. For this reason, it
would seem useful to plot the Lorenz curve upon triangular
or tri-axial ordinates, by omitting the useless half of the square
chart-field, and to make the stright-line diagonal the base of
the chart in order to emphasize the deviation of the curve
from the straight-line diagonal. A second feature of the
LORENZ CURVES
365
Lorenz curve is that the lack of similarity between the two
terminal parts or “tails” of the curve, indicates what statis-
ticians call skewness in the data. In these ways, this form of
chart is a useful tool for the technician, and yields an intelli-
gible message of details in which he is interested and which
will escape the lay reader. Primarily, however, the Lorenz
curve is popular — the one and only way of making an inter-
esting picture of a frequency distribution for the average man
and of bringing strongly home to him the practical aspects
of the frequency distribution presented.
PART III. RATE-OF-CHANGE ANALYSIS
Chapter XXXI
THE GENEALOGY OF NUMBERS
The first mathematical operation in the world was piobably
more difficult for its discoverer, than the most complicated
mathematical processes are for us today. Prehistoric man was
able to master that initial operation and hence you are con-
fronted with the disconcerting question, ‘‘are you intellectually
weaker than the cave man, the stone-age man and the iron-age
man?'^ If you admit the charge, or if you have already mas-
tered higher mathematics, you should skip this chapter and
continue in the straight and narrow road of charting, for in the
first case you will not understand it and in the second you
will not need it. The chapter is by way of being a comic inter-
lude in which the reader is invited to wander down a by-way
into pure mathematics, which will give him a theoretical
understanding of the charts which follow. For a practical
working knowledge of them, this theoretical understanding
is not needed, but it will be a source of abiding satisfaction to
him and will incidentally raise his batting average against
charting errors and mistakes.
The first mathematical operation was that of counting off
or numbering. That is to say, standing in the middle of a
road, you walked forward and your first step was your “first^’
step, your next was your “second'^ step, your next was your
“third’^ step, and so on. The implements are called ordinals
(“first, “second, “third, etc.). The result of this counting
off or numbering is to give you the number of steps you have
taken, that is, measurement or mensuration. The measure-
ment comes in the form of what is called a cardinal number
(“one,’’ “two,” “three,” and so on). Hence note that you
have the following:
Materials: Distinct items
Operation: Counting off or numbering
Result: Ordinal numbers
366
THE GENEALOGY OF NUMBERS
367
Materials: Ordinals
Operation: Measurement or mensuration
Result: Cardinal numbers
Several centuries may have passed before another bright
young chap came along who was a little less hairy than his
ancestors and had a little higher forehead. He discovered
that if he walked five miles one day and five the next, it was
the same as walking ten miles in all, the interesting thing being
that the same two cardinals always made the same third car-
dinal. This made possible the great and fundamental law
upon which our civilization is said to rest, that two and two
make four. The operation is known as ^‘addition/^ It is a
sort of multiple measuring. Its result is a ‘‘sum.’’ An in-
verse operation exists which is called “subtraction,” the result
of which is called a “difference.” In this inverse operation we
first meet with what are called “negative numbers” giving
rise to a conception that numbers can sometimes be either
“positive” or “negative.” And in this inverse operation we
must distinguish the two numbers operated on, calling the first
the “minuend” and the second the “subtrahend.” Now note
that you have the following:
Materials:
1st material
2nd material
Operation:
Result:
Direct
Terms
Increment
Addition
Sum
Inverse
Terms
Minuend
Subtrahend
Subtraction
Difference
Again many centuries elapse before the third act. This
time the inventor discovers that if he walks five miles every
day for five days, it is the same as walking twenty-five miles
in all,' and that no matter how often he repeats the operation
the result will always be the same for each two given numbers.
From this we discover the law that two times two make four,
and with it the multiplication table. The process is called
multiplication and is a sort of multiple adding. Most of our
so-called multiplying machines are built on this principle, the
operator simply turning the crank of the machine often enough
to add the quantity the right number of times. The reverse
process is called ‘division” and is a sort of multiple subtrac-
tion, The result of “multiplication” is called a “product,” of
division a “quotient.” And the reverse process, division, gives
368
CHARTS AND GRAPHS
rise to a new type of number called a “fraction/’ showing us
that numbers can sometimes be “integrals” or whole numbers
and sometimes “fractions” or part numbers. And remembering
the negative numbers we met, we also find negative products
and -quotients or fractions. Note now that we have the fol-
lowing :
Direct
Inverse
Materials:
F actors
F actors
1st material
Multiplicand
Dividend
2nd material
Multiplier
Divisor
Operation:
Multiplication
Division
Result:
Product
Quotient
Again a long time passes. The fourth act begins. Some
one says: A five-mile distance walked by five persons, total
walking, 25 miles (in which 5 is used twice as a factor), on 5
-different days, total 125 miles (in which 5 is used three times
as a factor), in five different cities, total 625 miles (in which 5
is used four times as a factor) and in five different countries,
total 3125 miles (in which 5 is used 5 times as a factor). Now,
says he, instead of writing“5X5X 5X5X5” why not write 5^
and be done with it. His invention, you see, is clearly one of
notation. His process is called “raising to a power,” his result
being a “power” of the original number. It is a sort of multiple
multiplication. The inverse operation is called “reducing to
a root” the result being a root of the original number. It is a
sort of multiple division. We use the method- when we say
that the square (or second power) of two is four. The little
number up in the corner is called the exponent. When it is in
the righthand corner it signifies raising to a power, and when
on the lefthand side in a radical sign it signifies the inverse
operation of extracting a root. Again the raising to a frac-
tional power also signifies the inverse process. And in the in-
verse process we meet with the square root of negative num-
bep, which we call “surds” or “irrational numbers.” And we
^ also find negative exponents. Now note that you have:
Direct Inverse '
Operation: Involution Evolution
Materials: 1st Number Number
2nd Exponent Fractional exponent
Power Root
Result:
THE GENEALOGY OF NUMBERS
3h
Here, the author, in the role of stage manager, must step
out in front of the curtain, with a little speech of apology.
The play would progress better had not the playwrights, that
is the mathmeticians, been badly put to it to find new names
and symbols for their operations. They have progressed
bravely up to this point. Thus reviewing their work you find:
Counting off: 1st, 2nd, 3rd, 4th, 5th, etc., gives us Measurement;
1, 2, 3, 4, 5, etc.
Multiple Measuring: 1, 2, 3, 4, 5, and 1, 2, 3, 4, 5, gives us
Addition; 5 -|-5 =10.
Multiple Addition: 5 + S + S + 5 + S gives us Multiplication;
5x5 =25.
Multiple Multiplication: 5xSxSx5x5 gives us an Involu-
tion; 5^ = 3125.
So far, the}^ have given us a marvelous system for the easy
notation of their ideas. You will notice that the phrase 5''^ is
a highly compressed expression, which would otherwise have
to be written 5 X 5 X 5 . . (to 5 times) or 5 +5 +5 4-5 +5 . .
(to 625 times) or 1 4-1 4-1 4-1 4-1 . . . (to 3125 times). But
we warn you that this simplicity is at an end. Examine the ex-
pression, 3125 =5^. Substitute for it the general algebraic ex-
pression A = This describes A as the C-th power of B,
From it you can readily derive the expression for 5, as follows:
c
B = VA^ that is B is the C-th root of A, But they have no
convenient symbol for C, the exponent of the power to which
B must be raised to equal A, and they can only give you a
cumbersome word by which you can describe C as — but wait
and see.
The fifth act, with which, so far as we are concerned, the
play should end happily, opens with a young man who discovers
that the fifth power of five is the same thing as multiplying to-,
getherthe second and third powers of five, and that, in general,
to multiply two powers of the same number together you need,
merely add their exponents, thus B"^ X B"^ = Likewise,,
to divide a power by another power of the same number, you
need merely subtract their exponents, thus B^ ^ B'^ =
Whereupon he promptly says, let us change all numbers in the
world into powers of one common and universal base number
and then we shall be able to substitute for the lengthy tedious
process of multiplication and division, the simple and easy
process of addition and subtraction. Instead of multiplying
370 CHARTS AND GRAPHS
together a long series of large numbers we would only need
to add their corresponding exponents of this universal base.
The discovery was in the nature of a miracle. To add expo-
nents instead of multiplying powers! The process has been
accounted one of the nine wonders of the world. And it’s as
easy as falling olF a log!
Then this excellent discoverer has to spoil his work by
using a long and terrifying name for his process. For he calls
it logarithmation. He calls his exponents of the common base,
logarithms to that base. He calls his table of exponents a table of
logarithms. He cannot think up a new symbol and when
asked what c is in the equation, A = he writes:
c^log^A
(This is read, “c equals log A to base B’\) For he abbreviates
his long word logarithm by the short word log. But it is no
use. The public has decided that the use of logarithms is not
as easy as falling off a log. All of which goes to show that
there is something in a name after all. The public fell off the
logs long ago and has been off them ever since. After this
unhappy denouement, we introduce the following pageant,
as additional entertainment to an audience which has sat
faithfully through five tedious educational acts.i
Marshal before your eyes the countless myriads of num-
bers known to man. (For the sake of simplicity consider the
whole numbers only, forgetting for the moment the fractions.)
Arranged in single file from zero out into infinity, it would
take forever for their procession to pass. For there is literally
no end to them. Marching by at the rate of one at every
tick of the clock, the first ten thousand would pass in an hour.
And marching day and night, after four days one million would
appear. But it would be ten years before the first number of
ten digits, the first billion number, comes into view. And as
to the trillion, that would not yet have appeared if the parade
had begun before the pyramids were built. Yet the trillion
is no longer a stranger to financial circles and is a poor small
thing in the world of science.
Now hovering over the shoulder of every one of these num-
bers the close observer might discover its spiritual counterpart,
its soul. Subject this soul to close analysis and you will fi nd
* The foregoing text has been largely modelled after the excellent introductory
chapter in “Engineering Mathematics’* by Charles P. Steinmetz.
THE GENEALOGY OF NUMBERS
371
it is the exponent which will raise some common universal
base-number to the value of the number itself, and since our
numbers are arranged on the decimal system, the most con-
venient base figure for the exponents or souls is the number
ten. From this we may easily identify the souls of all powers
of ten. Thus it is easy to see that the soul of ten itself is 1,
since ten is the first power of ten. It is easy to see that the
soul of one hundred is 2, since it is the second power of ten
(that is, the base number, ten, must be taken twice as a factor
to give us the number one hundred). It is easy to see that the
soul of one thousand is 3 ; that the soul of one million, for ex-
ample, is 6; and so on. Indeed we quickly discover that every
whole or integral soul belongs to an even power of ten and coin-
cides with the number of ciphers between the initial digit one
and the decimal point. In short, the soul tells us the position
of the decimal point.
Going back into small numbers and fractions it is equally
simple. The soul of one, for instance, is 0, for the zero power
of any number, including the base ten, is one. Notice that the
soul still tells us the position of the decimal point, for the
latter is immediately beside the initial digit, without any inter-
vening digits. Now what is the soul of one tenth? Obviously
it is -1, for as you know, to convert a denominator into a
numerator we need merely change the sign of its exponent, and
10
=io-‘.
Likewise the soul of one-hundredth is -2, for
100
1
— = 10 '^
102 •
The soul of one-thousandth is
. 3 , of one ten-
thousandth is -4, of one-millionth is -6, and so on. And if
we write these fractions, one-tenth^ one-hundredthy one-thou-
sandth, one ten-thousandth, or one-millionth, as, respectively,
‘^000,r^ or ^^000,001,^’ we shall see that
their souls, namely -1, -2, -3, -4 or -6, tell us again the
positions of the decimal points. Only this time, since the
decimal point has been moved backwards, that is, to the left of
its position beside the first significant digit (initial ciphers are
not called significant), the soul has become negative. A
quaint and convenient fact, that the soul always tells, by its
sign and whole or integral part, the precise position of the
decimal point in a number.
CHARTS AND GRAPHS
372
But what of the souls of numbers lying between the even
powers of ten ? There are eight numbers between one and teUj
eighty-nine between ten and one hundred^ and many, many
more between the higher powers. They too have souls, but
it is clear that they cannot have even whole or integral souls,
for these belong to the powers themselves. The numbers too,
three^ four and so on, for example, lie between one and ten; so
they must have souls between 0 and 1. We easily conclude,
therefore, that their souls must be fractions somewhere be-
tween 0.0000 and 1.0000. That this is correct we can quickly
demonstrate. Consider ike square root of ten, a number, as you
know, a little greater than three. It is obvious that its soul
must be since the square root of ten is the one-half power of
^ 2
ten, that is VIO—IO^. Hence for the square root of ten we
have a soul 0.5000, lying as you see between 0.0000 and 1.0000.
Mathematicians have figured out to many places the souls of
other numbers, that of three (which is but a little less than the
square root of ten) being to four places 0.4998; that of two being
0.3010. Remember these two and you will always be able to
reconstruct the souls of almost all other numbers without as-
sistance. In short, the souls of all numbers other than powers
of ten are not integral, but are fractional.
Returning to the parade, let us call a halt to the intermin-
able thing and hold a grand review of the numbers, marshalling
them according to their significant digits. In the entire bat-
talion of numbers there will then be but nine regiments, each
led by one of the significant digits, one, tzuOy three^ four^ five^ six,
seven, eight, or nine. Each regiment will again be composed
of ten companies in which the second digit is one of the ten
numerals, zero to nine. Each company will be divided into
ten platoons in which the third digit likewise varies,, each
platoon into ten squads whose fourth digits vary, and each
squad will be. similarly divided into ten subdivisions. This
subdivision could proceed indefinitely. You will notice that we
have here disregarded entirely the position of the decimal
point. Now the interesting thing about this arrangement is
the fractional parts of the souls. For the souls would never
repeat themselves in this review, but there would be one
ftactional part of- a soul assigned to each file or succession of
significant digits, and belonging to that particular file al-
THE GENEALOGY OF NUMBERS
373
ways, regardless of changes in the position of the decimal
point. The integral parts of the souls would indeed change
with the change of the decimal point, but not the fraction.
Thus glancing down the file ‘‘200000,’’ we find that the soul
of two (“2.00000”) is 0.3010, that the soul of twenty
(“20.0000”) is 1.3010, that the soul of two hundred (“200.000”
is 2.3010, that the soul of two thousand (“2,000.00”) is 3.3010,
and so on. In short, while the integral parts of souls are the
same for similar positions of the decimal point, the frac-
tional parts of the soul are the same for similar successions of
(significant) digits in numbers.
But this glimpse of the souls of numbers must come to a
close. From now on in this book (and in all other books), you
will meet them again only under the prosaic names of logs or
logarithms, or more precisely, common or Briggsian logarithms.^
When logarithms are used, the natural numbers for which
they stand are sometimes called anti-logarithms. The frac-
tional part of the logarithm is called the “mantissa” and be-
cause it is the same for all similar combinations of natural
numbers or significant digits, it forms the body of the table
of common logarithms. The integral or whole part of the
logarithm is called the “characteristic,” and since it records
the position of the decimal point is not shown in logarithm
tables but is left to be determined by inspection. With the
information dispensed in this chapter, in as heavily sugar-coated
pellets as we could provide, you are prepared to meet, master
and make use of any logarithm which strays your way as if
it had been your life-long servant — ^no, no, much better than
that!
In this chapter we shall go no further into the uses of
logarithms. They shall sit up and perform for us through the
major part of the rest of this book. We merely repeat, for
your lasting remembrance (and don’t ever forget it) their
fundamental relations:
log A+\og 5 = log {AxE)
log A -log 5 = log {A -^B)
^ “Logarithms were invented and a table published in 1614 by John Napier of
Scotland; but the kind now chiefly in use proposed by his contemporary, Henry
Briggs, professor of geometry in Gresham College in London .’* — Century Dictionary.
374
CHARTS AND GRAPHS
N
0
1
2
3
4
5
6
7
8
9
\\
12345
10
0000
0043
0086
0128
0170
0212
0253
0294
0334
0374
4 a 12 17 21
11
0414
0453
0492
0531
0569
0607
0645
0682
0719
0755
4 8 n 15 19
12
0792
0828
0864
0899
0934
0969
1004
1038
1072
1106
3 7.10 14 17
13
1139
1173
1206
1239
1271
1303
1335
1367
1399
1430
3 6 U' 13 16
14
1461
1492
1523
1553
1584
1614
1644
1673
1703
1732
36 3 12 15
15
1761
1790
1818
1847
1875
1903
1931
1959
1987
2014
3. e. 8 1 1 H
16
2041
2068
2095
2122
2148
2175
2201
2227
2253
2279
3 5 8 11 n
17
2304
2330
2355
2380
2405
2430
2455
2480
2504
2529
2 5. 7 10 12
18
2553
2577
2601
2625
2648
2672
2695
2718
2742
2765
2 5. 7. 9 12
19
2788
2810
2833
2856
2878
2900
2923
2945
2967
2989
2 4 7 911
20
3010
3032
3054
3075
3096
3118
3139
3160
3181
3201
2. 4 6. 8 1 1
21
3222
3243
3263
3284
3304
3324
3345
3365
3385
3404
2 4 6 8 10
22
3424
3444
3464
3483
3502
3522
3541
3560
3579
3598
2. 4- 6. 8 10
23
3617
3636
3655
3674
3692
3711
3729
3747
3766
3784
2 4. 5 7 si
24
3802
3820
3838
3856
3874
3892
3909
3927
3945
3962
2- 4. 5 7. 9
25
3979
3997
4014
4031
4048
4065
4082
4099
4116
4133
2 5- 5 7. 9
26
4150
4166
4183
4200
4216
4232
4249
4265
4281
4298
2 3 5 7 9
27
4314
4330
4346
4362
4378
4393
4409
4425
4440
4456
2* 3 5. 6 8
28
4472
4437
4502
4518
4533
4548
4564
4579
4594
4609
2- 3- 5- 6 8
29
4624
4639
4654
4669
4683
4698
4713 4728
4742
4757
1. 3. 4. 6. tj
30
4771
4786
4800
4814
4329
4843
4857
4871
4886
4900
1 3. 4. 6. :
31
4914
4928
4942
4955
4969
4983
4997
5011
5024
5038
1. 3. 4. 6* 7
32
5051
5065
5079
5092
5105
5119
5132
5145
5159
5172
1 3. 4 5 7 j
33
5185
5198
5211
5224
5237
5250
5263
5276
5289
5302
1. 3* 4. 5. 6!
34
5315
5328
5340
5353
5366
5378
5391
5403
5416
5428
1 3 4. 5. ei
35
54.41
5453
5465
5478
5490
5502
5514
5527
5539
5551
1-2 4 5 6
38
5563
5575
5587
5599
5611
5623
5635
5647
5658
5670
I 2- 4. 5> 6
37
5682
5694
5705
5717
5729
5740
5752
5763
5775
5786
1 2. 3 5^ 6
38
6798
5809
5821
5832
5843
5855
5866
5877
5888
5899
1. a. 3* 5'. 6
39
5911
5922'
5933
5944
5955
5966
6977
5988, 5999 60I0
1-2. 3.4 6
40
6021
6031
6042
6053
6064
6075 6085
6096
6107
6117
1 2. 3. 4. 5
41
6128
6138
6149
6160
6170
6180 6191
6201
6212
6222
1-2. 3. 4. 5
42
6232
6243
6253
6263
6274
6284
6294
6304
6314
6325
1-2. 3* 4. 5
43
6335
6345
6355
6365
6375
6385
6395
6405
6415
6425
2. 3* 4. 5
44
6435
6444
6464
6464
6474
6484
6493
6503
66|13
6622
1- 2* 3* 4 si
45
6532
6542
6651
6561
6571
6580
6590 '6599
6609
6618
1 2. 3- 4. 5
46
6628
6637
8646
6656
6665
6675
6684
6693
6702
6712
1 2. 3. 4 5
47
6721
6730
6739
6749
6758
6767
6776
6785
6794
6803
1 2 3. 4. 5
48
6812
6821
6830
6839
6848
,6857
6866
6876
6884
6893
1 2. 3 4. 4
49
6902
6911
6920
6928
6937
6946
69'55
6964
6972
6981
1. 2. 3-4. 4
50
6990
6998
7007
7016
7024
7033
7042
7050
7059
7067
12. 3- 3t 4
51
7076 7084
7093
7101
7110
7118
7128 7135' 7143 7162
1/2. 3s 3. 4
52
7160 7168
7177
7185
7193
7202
7210
7218
7226
7235
12 2. 3 4
53
7243
7251
7259
7267
7275
7284
7292
7300 7308
7318
1. 2. 2 3. 4
54
7324
733'2
7340
7348
7350
7364
7372
7380
7388 *7396 1
1, 2. 2. 3. 4
Fig. 327. Table of Logarithms, t-5.
THE GENEALOGY OF NUMBERS
55
7404
7412
7419
7427
7435
56
7482
7490
7497
7505
7513
67
7559
7566
7574
7582
7589
58
7634
7642
7649
7657
7664
59
7709 7716
7723
7731
7738
60
7782
7789
779S
7803
7810
61
7853
7860
7868
78'75
7882
62
7024
7931
7938
7945
7952
63
7993
8000
8007
8014
8021
64
8062
8069
8075
8082
8089
65
8129
8138
8142
8149
8156
66
8195
8202
8209
8215
Z!i22
67
8261
8267
8274. 8280
8287
68
8325
8331
8338
8344
8-351
69
8388
8395
8401
8407
8414
70
8451
8457
8463
8470 84761
71
8513
8519
8525
8631
8537
72
8573
8579
8585
8591
8597
73
8633
8639
8645
8651
8657
74
8692
8698
8704
8710
8716
75
8751
8756
8762
8768
8774
78
8808
8814
8820
8825
8831
77
8865
8871
8876
8882
8887
78
8921
8927
8932
8938
8943
79
8976
8982
8987
8993
8998
80
9031
9038
9042
9047
9053
81
9085
9090
9096
9101
9106
88
9138
9143
9149
9154
9159
83
9191
9X96
9201
9208
9212
84
9243 9248
9253
9258
9263
85
9294 9299 9304 9309
9315
86
9345
9350
9355
9360
9365
87
9395
9400 9405
9410
9415
88
9445
9450
9455
9460
9465
86
9494
9499
9504
9509
9513
90
9542
9547
9552
9557
9562
91
9590
9595
9600
9605
9609
92
9638
9643
9647
9652
9657
93
9685
9689
9694
9699
9703
94
9731
9736
9741
9745
9750
95
9777
9782
9786
9791
9795
96
9823
9827
9832
9836
9841
97
9868
9872
9877
9881
9886
98
9912
9917
9921
9926
99S0
99
9956
9961
9965
9969
9974
r
7443
7451
7459
7466
7474
7520
7528
7536
7543
7551
.7597
7604
7612
7619
7627
7672
7679
7686
7694
7701
7745
7752
7760
7767
7774
7818
7825
7832
7939
7846
7889
7896
7903
7910
7917
7959
7966
7973
7980
7987
8028
803^
8041
8048
8035
8096
8102
8109
8116
8122
8162
8169
8176
8182
8189
8228
8235
8241
8248
8254
8293
8299
8306
8312
8319
8357
8363
8370
8376
8382
8420
8426
8432
8439
8445
8482
8488
8494
8500
8506
8543
8549
8555
8561
8567
8803
8609
8615
8621
8627
8663
8669
8675
8681
8686
8722
8727
8733
8739
8745
8779
8785
8791
8797
8802
8837
8842
8848
8854
8859
8893
8899
8904
8910
8915
8949
8954
8960
8965
8971
9004
9009
9015
9020
9025
9058
9063
9069
9074
9079
9112
9117
9122
9128
9133
9165
9170
9175
9180
9186
9217
9222
9227
9232
9238
9269
9274
9279
9284
9289
9320
9325
9330
9335
9340
9370
9375
9380
9385
9390
9420
9425
9430
9435
9440
9469
9474
9479
9484
9489
9518
9523
9528
9533
9538
9566
9571
9576
9581
9586
9614
9619
9624
9628
9633
9661
9666
9671
9675
9680
9708
9713
9717
9722
9727
9754
9759
9763
9768
9773
Fig. 328. Table of Logarithms, 5-9
376
CHARTS AND GRAPHS
A device which will do all this is worth knowing.2 In the
language of valedictorians, we commend to your early atten-
tion a small table of logarithms and many pleasant hours of
easier computing therewith.
2 From logs, that is, the logarithms of numbers, it is but a simple step to proceed
to loglogs, that is, the logarithms of logarithms. Consider the phrase in which
A and B are any values we wish, such as 29.37 and 43.921. We can write log
log A. This reduces the involution to a mere matter of multiplying B into the log
of A, But if the multiplication be tedious, as with the values first instanced, it will
be simpler to write
log (log A^) —log {B log A)
=log .^-j-loglog A
and proceed by addition. The loglog of a number is the logarithm of its logarithm,
and is found in the log tables by treating the logarithm as a number.
Chapter XXXII
THE LAW OF ORGANIC GROWTH
The law of organic growth, as it is called, is well-nigh as
important to the practical business man as the law of cause
and effect, but is unfortunately much less understood. The
chart papers and methods discussed in this section of the
book are designed to interpret statistics in the light of this law.
To the uninitiated their construction remains a mystery, but
to those who know the law which is the key to their meaning
they are so valuable as to eclipse and almost to obviate all
other chart methods. The law relates to the way in which a
large majority of natural organic forces have been found to
grow or change. It prescribes or defines the manner in which
this growth or change will take place. The law is that, at
regular intervals of time, each new value will be a constant
percentage of the immediately preceding value.
This feature of a constant relation between successive
items in a series marks what mathematicians call a progression.
There are several kinds of progressions, only one of which
follows the law of organic growth. By far the simplest form
of progression is the one called arithmetical. In the arith-
metical series or progression, each item differs from the pre-
ceding item by a constant amount (quantity, difference or in-
crement). The series progresses from item to item either by
addition or by subtraction of this amount. For example, in
the series, 1, 2, 3, 4, 5, 6, . . ., the constant increment is +1.
In the series 4, 3, 2, 1, 0, - 1, -2, . . ., the constant is ~1. In
business, the familiar instance of the arithmetical progression
is the accumulation of simple interest.
Another and a very different series or progression is the one
called geometrical. In a geometrical progression, each item
differs from the preceding item by a constant ratio (rate, per-
centage, factor, multiplier, or divisor). The series progresses
.from item to item by multiplication or division by this con-
377
378
CHARTS AND GRAPHS
stant ratio. For example, in the series, 1, 2, 4, 8, 16, 32, 64,
. * . ,the constant ratio or factor is 2. In the series 4, 2, 1,
h h h • • * ^he constant is In business, a familiar
instance of the geometrical progression is the accumulation of
compound interest. And it is the geometrical progression
which the law of organic growth prescribes.
It is interesting to study these two types of progression,
for they are the gist of the distinction between the curve
charts which we have so far considered and the curve charts
to which we are coming. In the first place let us consider the
relative speed of these progressions. Compare the series 1, 2,
3,4, 5, 6, . • with the series, 1, 2,4, 8, 16, 32, 64, . . ., and
you will see that the geometric progression rapidly outruns
the arithmetical one. These series have begun at unity and
progressed by a 100% increase, which is a fairly rapid rate of
increase. But the acceleration, from the arithmetical point
of view, of the geometrical series will still be evident if we
take A slower rate of increase such as 10%. Starting at unity,
the arithmetical series will be 1, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7,
• . . while the geometrical series will be 1, 1.1, 1-21, 1.33,
1.47, 1.62, 1.78, 1.96. ... Of course either of the two types
of progressions can begin with any item and increase at any
rate, but from any point you wish to choose, if the rates are
the same for the two series, the geometrical progression will
always increase more rapidly than the arithmetical one.
On the other hand, in the decreasing or diminishing direc-
tion, the arithmetical progression will leave the geometrical
one behind. Compare the series, 2, 1, 0, —1, -2, -3, . . .
with the series 2, 1, f, and this will be evi-
dent to you. And here we come to an important distinction
between the two series, namely that while the arithmetical
series can reach zero and pass into negative values, the geo-
metrical series can never reach zero at all. In the example
just given, the arithmetical series diminishes by subtracting
I of the value of its first item and quickly passes zero, but
the geometrical series diminishes by division by 2, and gives
no indication of ever reaching the value of zero. Now we
could have made the geometrical series begin with any other
positive value and decrease at any other rate we please, but
it still would be impossible for us to bring the geometrical
progression down to zero. We can, by constantly diminishing
it, that is, by repeatedly dividing its last item, bring it as
THE LAW OF ORGANIC GROWTH 379
close to zero as we please, without ever succeeding In entirely
wiping it away. In mathematical language, zero is the in-
finitesimal limit of the geometrical progression.
Malthus made popular the distinction between these two
kinds of progression with his theory that while the population
of the world increased geometrically, the wealth of the world
increased only arithmetically and hence soon limited the wel-
fare of the population. But his theory has been proved to be
false, the wealth of the world appearing to increase geomet-
rically, though sometimes at a slower rate than the popula-
tion. Indeed an attempt has recently been made ^ to invest
mankind with a peculiar and to some extent exclusive power
of geometrical progression, both in mental and physical accom-
plishments, a theory to which the work of Professor Ogburn^
in suggesting something like a geometrical progression of all
that may be called human civilization, bears important con-
firmation. While it may not be so certain that the power of
geometrical development is exclusively a property of human
individuals, not shared by individual animals and plants, it
is safe to say that human accomplishments, including business
enterprises, are as often subject to the law of organic growth
as are natural forces.
The law of organic growth therefore is the proper criterion
for the business man in judging the development of a business,
as it is for the economist in his study of industrial and socio-
logical records. In applying this criterion, we must forget
(for the time being at least) the amount of increase in our
business from year to year and center our attention upon its
rate of increase. What are the year-to-year percentages of
increase.? If the 1911 sales were 10% larger than the 1910
sales and the 1912 sales were again 10% greater than the 1911
sales, the business has increased in accordance with the law of
organic growth. If the 1913 sales were again 10% greater than
those in 1912, the increase has still followed the law. If the
1913 sales were only 8% greater than those in 1912, there
has been from the point of view of the law, a definite slowing
down, or falling off in the rate of growth, which must either
be explained by conditions outside the control of the company,
such as a general business depression, or is a harbinger cf ill
^ Cf. Korzybski, Alfred, Manhood of Humanity, E. P. Dutton k Co.,NewYork, 1921.
^ Cf. Ogburn, William Fielding, Social Cha^ige, B. W. Huebsch, Inc., New York,
1922.
380 CHARTS AND GRAPHS
omen which should call for almost as careful consideration by
the directors as if there had been an absolute loss. On the
other hand, if the 1913 sales were 15% greater than those in
1912 and the event is not to be explained by forces outside
of the control of the company, there is reason for far-sighted
rejoicing and thanksgiving among the directors.
That the results of the use of this criterion are radically
different from the results reached by a study of the amount of
change, is evident from the fact that the former may at times
be directly contradictory to the latter. Let us suppose that
in its first year the gross sales of the house amount to ^50,000,
and in the second year to $100,000, that is, there has been an
increase of $50,000, or 100%. If in the third year, sales
amount to $160,000, obviously the amount of increase has gone
up from $50,000 to $60,000, but the rate of increase has fallen
from 100% to 60%. If in the fourth year sales amount to
$225,000, the amount of annual increase has again risen to
$65,000 but the rate of annual increase has fallen to about
40%. If in the fifth year sales amount to $300,000, the
amount of annual increase has again risen, this time to $75,000,
and the rate of annual increase has again fallen, this time to
33%. This fictitious example makes clear how illusory would
be any conclusion based wholly upon the amount of change
from year to year and how Important it is, that the annual
rate of change should be watched, that is, that the records
should be studied in the light of the law of organic growth.
As a matter of fact, few business houses follow closely the
law of organic growth for any considerable period of time. Or
perhaps it would be better to say that though operating under
the law of organic growth, they fail to maintain a constant rate
of change. For it is certain that they operate under this law
rather than under any law of arithmetical progression. Theo-
retically, perhaps, given constantly similar external conditions
and internal efficiency, the growth of a business house would
conform to a geometrical series and illustrate perfectly the
law. But as a matter of fact, the individual business house is
at the mercy of a large number of external forces which lie
outside of its control and do not remain constant but are ever
changing, and the records of its growth therefore show a great
amount of the play of what we might call chance variation.
Business men are accustomed to thinking, in some fields, of a
very definite saturation point in their markets. Of course when
THE LAW OF ORGANIC GROWTH
381
such a point is approached, it becomes a limit which will neces-
sitate a slowing up of the rate of increase, in spite of otherwise
equal conditions, which would have favored a strict adherence
to the geometrical progression. These individual variations
and any approach to limiting points do not invalidate the law
of organic growth, nor do they diminish its value as a criterion
of business success. They are separate and additional forces
imposed upon the development * of the individual business,
their co-action with the law of organic growth determining the
fluctuating records of the house.
In entire industries, or in large aggregates of individual
business records, the adherence to the law of organic growth
is much closer and the operation of the law easily seen. In
the last two decades, the automotive industry has afforded a
spectacular illustration of a geometrical progression with a
very rapid rate of increase, although a decline in this rate in
recent years is terrifying manufacturers with visions of the
approach of a potential saturation point in their domestic
market.^ Two other recent industries, whose entire history
can be covered in the last twenty years, show similarly close
adherence to the law, both the phonograph and the moving-
picture industries having grown by leaps and bounds, which
when analyzed in this way become surprisingly regular and
uniform. The operation of the law of organic growth in bus-
iness and economic affairs is even more rigid in national and
world-wide records.
' ^ There is another curve which sometimes fits economic data better than the log-
arithmic curve, namely the Gompertz curve. This is dealt with more fully later on,
in the chapter on Special Projections.
Chapter XXXIII
RATE-OF^CHANGE ANALYSIS
To subject a business record or the events in any other
phenomenon to analysis according to the law of organic
growth is not really a difl&cult problem. It implies of course
a comparison of the rate of change or ratios between each two
successive items in a series, and these successive ratios are the
successive quotients obtained by dividing each item by the
item immediately preceding it. This method of successive
divisions would, however, be tedious even for the shortest
series, and when applied to the wholesale analysis of a large
body of statistics, such as (in the individual business house) the
records of various lines and articles, or (in economics) the
statistics for many industries or sociological developments,
would indeed mount up to a forbidding and costly task. Mere
inspection of the data, from which we could at once detect an
arithmetical progression or recognize the failure of a given
series of figures to conform to an arithmetical progression, will
not often suffice to dig out the geometrical progression.
Indeed, a fairly close approximation to a geometrical series
may be so completely veiled in the figures that it passes un-
noticed through close and even expert inspection of the data.
The question, therefore, is can the geometrical series be made
as apparent as the arithmetical one.^ And how can the failure
or deviation of a series of data from a straight geometrical
progression be as easily measured as its deviation from an
arithmetical one?
Between the two types of progressions, arithmetical and
geometrical, there is a curious inter-relation which it is well
to master. Let us examine again the series 1, 2, 4, 8, 16, 32,
64, . . . With the exception of the first item in this series,
it is evident that all the items are merely powers of 2 and
since, as you know, 1 is merely the 0 power of any number,
we may include it in the series calling it the 0 power of 2.
382
RATE-OF-CHANGE ANALYSIS
383
Therefore we can rewrite this series as follows: 2^, 2^, 3^, 2^, 2^,
2^ 26.... Now examine these exponents and you will find
that they form an arithmetical progression, 0, 1, 2, 3, 4, Sy 6.
... In short, we find that if a geometrical series be rewritten
as a series of the powers of a single quantity (the constant
multiplier or rate of change), the exponents of these powers
will form an arithmetical progression. This is a rule of general
application and might be used as a means of defining the
geometrical series.
Here we come again to logarithms. For a logarithm, you
will remember, is merely the exponent by which a common or
universal base figure can be raised to a given value. In re-
writing the series 1, 2, 4, 8, 16, 32, 64, ... as the series
2^, 2b 22, 2^, 2^, 26, 26, . . . we are obviously using 2 as a
common or universal base for the series and the exponents,
0, 1, 2, 3, 4, 5, 6, which will raise this base figure 2, to the
value of the items in the original series may be called the
logarithms (to base 2) of the items in the original series.
Since 1=2^ it is obvious that 0 = log2l, since 2=2b clearly
1 =:log22, and since 64 =2^, 6 = log264. In short, in turning the
geometrical series into a series of the successive powers of a
common base figure, we find ourselves writing as exponents
of these powers, the logarithms of the original series (to the
base figure which was used as the root of these powers). And
therefore we may say that the logarithms of the items in a
geometrical series will form in themselves an arithmetical pro-
gression. This is indeed a very usual definition of the geo-
metrical progression, namely that it is composed of items whose
logarithms form an arithmetical progression.
Logarithms can be taken to any common base figure we
desire. For example, the same geometrical progression, 1, 2,
4, 8, 16, 32, 64 . . . may be, if we wish, written as a series of
the powers of 4, as follows : 4b 4:1^ 4b 42, 4®/ 2, 4^. . . . Or
that same series can be written as a series of the powers of
any other number provided we but care to do the necessary
calculating. Now as you know, logarithms are ordinarily
taken to the base figure 10 and not to the base figure 2 or 4 as
in the above examples. The reason for this is that our numbers
are arranged upon a decimal system and by taking the base
figure as 10 we are able to make the integral part of the logar-
ithm (characteristic) a mere record of the position of the
decimal point in the original number, and we are able to make
384
CHARTS AND GRAPHS
the fractional part of the logarithm (mantissa) the same for
all similar succession of similar digits. Suppose therefore that
we adopt 10 as the quantity whose powers we wish to sub-
stitute for the original series. In this case we rewrite the
series, 1, 2, 4, 8, 16, 32, 64 ... as the series 10^, 10^-^^^^,
10*^ 10^ 10^ . . . Again we find these expo-
nents or logarithms, 0.0000, 0.3010, 0.6020, 0.9030, 1.2040,
1.5050 . . . , forming an arithmetical series.
The inter-relation between the arithmetical and geometrical
series is therefore such that in a sense, both series may be
spoken of as arithmetical, the former being arithmetical in its
original form and the latter becoming arithmetical when its
logs are used. For this reason the term logarithmic is often
used synonymously with the term geometrical to distinguish
the latter form of progression and its item-to-item changes,
from the progression which is truly arithmetical in its original
form. For though the two types of progression are made
similar by the substitution of logarithms for one of them, yet
it must be remembered that they are two radically different
things which must never be confused with each other. In
their original form or natural numbers the one progresses by
addition or subtraction and the other by multiplication or
division and there is a world of difference between them. That
they behave similarly when logarithms are substituted for one
of them, is chiefly due to the peculiar qualities of logarithms,
that by their use the process of addition or subtraction may
be substituted for the process of multiplication or division
(instead of dividing one number into another number to get
a quotient, you subtract the logarithm of the first from the
logarithm of the second and the difference is the logarithm
of the quotient).
In our analysis of our statistics in the light of the law of
organic growth, we can find a short cut, therefore, through
the use of logarithms. In other words we can turn the items
in our data into their corresponding logarithms (consulting
for this purpose a table of common logarithms) and then, by
comparing the logarithms, we can quickly discover any uni-
formity or constancy in their amount of change, and so easily
detect and measure the degree of adherence in the original data
to the geometrical series. The deviation of the series of logar-
ithms from an arithmetical progression with uniform amount
of change, is a measure of the deviation of the original series
RATE-OF^CHANGE ANALYSIS
385
of data from a geometrical progression with a uniform rate of
change. In practice, this method is very simple and it may
be regarded as a distinct labor-saving device in the careful
analysis of statistics.
In the chapter on Index Numbers, you will recall reading
that relative figures (that is, percentages) can be substituted
for an absolute series or series of original data. While these
relatives are ordinarily computed with a single item in the
series as the norm or 100%, yet they can be computed with
each item taken as a percentage of the item immediately pre-
ceding it. In this case the series is called a series of
chain-relatives or chain-percentages. Now it is precisely a
series of chain-percentages which the method of successive
divisions already mentioned gives us. But you will find that
the short-cut method of successive subtraction of logarithms
does not yield results in precisely the same form. For the
short-cut method gives us the logarithmic differences and
these differences are the logarithms of the percentages them-
selves. If therefore we desire to know the rate of change in
terms of its percentages of change (and not in terms of the
logarithms of its percentages of change) we must again consult
our logarithm tables, if we are using the short-cut method, and
convert the logarithmic differences back into the percentages
of change (by substituting for each difference its anti-logar-
ithm). As a matter of fact, however, the chain-relatives or
chain-percentages are, except for very popular purposes, not
ordinarily of sufficient importance to justify this additional
labor.
You will observe, however, that we have not yet reduced
the work of rate-of-change analysis of our statistics to the same
simple and easy steps as are found in an amount-of-change
analysis, for the use of logarithm tables and the substitution
of logarithms for the natural numbers requires, even with great
proficiency, a considerable amount of time and effort. And in
the wholesale analysis of a large body of statistics by this
method, we will not only find the work long and tedious but
we should also expect to find a large number of errors creeping
into the work which would be difficult to detect. The short-
cut method has, it is true, eliminated the more difficult proc-
esses of division (or multiplication), and in the lack of special
calculating machines, is a long step in labor saving, but the
rate-of-change analysis is not yet as simple as an amount-of-
386
CHARTS AND GRAPHS
change analysis. In the next chapter this final step will be
taken and through the simple use of the graphic method com-
bined with the use of logarithms, the work of substituting logs
for natural numbers will be eliminated and the rate-of-change
analysis made as simple as the amount-of-change analysis.
Organic, percentage, geometric or logarithmic change
(whichever name you prefer) is growth in which the rate or
ratio of change is uniform. Increment, difference or arithmetic
change is growth in which the amount or quantity of change
is uniform. The former naturally forms the basis of judgment
for the fluctuations of phenomena which cannot be negative,
that is, which must always be positive. The latter is fre-
quently the better basis of judgment for the fluctuations of
phenomena which can be zero and negative as well as positive.
In general it is perhaps best to study your data from both
points of view.i
^ Speaking of the amount-of-change curve, Professor Marshall says: “Its defects
are such that many statisticians seldom use it except for the purpose of popular
exposition, and for this purpose, I must confess, it has great dangers.” — Alfred
Marshall, On the Graphic Method of Statistics, Jubilee Volume of the Royal Statistical
Society, June 22-24, 1885, pp. 251-260.
Chapter XXXIV
RATE^OF-CHANGE SCALES
The rate-of-change curve chart affords in some t aspects the
most powerful analysis known of statistical data. An attempt
has been made in the last chapter to explain the general theory
of this chart method, but a real insight into its various uses
can only be obtained from a study of its applications. The
method is really nothmg more than the charting of the logar-
ithms of numbers in the place of charting the numbers them-
selves. A careful reading of the last chapter will doubtless
have already suggested this process to the student, and it only
remains to set forth the technique of charting logarithms.
Indeed, as will be seen, such simplified methods have been
developed that it is not necessary for one to understand logar-
ithms or be proficient in their use in order to benefit from this
chart. In the present chapter the development and construc-
tion of these simplified charts will accordingly be discussed with
the general principles covering the use of logarithms in the
charts.
Three methods are open to us in the plotting of logarithmic
curves. The first is the obvious one of substituting for the
items in a series to be plotted, the logarithms of those items.
We must consult a table of logarithms and for each item find
the logarithm and tabulate these logarithms in a column beside
our original series of data. Then on the plain co-ordinate
paper used for amount-of-change curves, in which the scales
are arithmetically projected (that is, the scale-figures at equal
distances form an arithmetical series), we must plot these
logarithms, and draw the curve through these plotted points.
The result, of course, will be a curve of the logarithms of our
original series, or, as we have called it, a logarithmic (or
^‘rate-of-change’’) curve. This curve will behave as a logar-
ithmic curve should and will tell us what we wish to learn
from the use of logarithms. The straight line, which always
387
388
CHARTS AND GRAPHS
PRICE OP POTATOES
Average Retail Price per Pound
United States
1913«*1920
(Source:- Btxreau of Labor Statistics}
3
o LOgar- OlOOlHtoiooiOi
Ck -t+ViTn COiOC-tOtOOC-O
lunm cvicoH-^J^toioior-
fU *•••*«•«
01 J><]OiOl>tOCQcOtO
tO*«j<tOtOl>CDCf>0
o>o>a>cs>o>o>a>S
Fig. 329. The Rate-of-change Curve — First Method.
This scale carries the logarithms, not the numbers
represents equal amounts of change and therefore depicts an
arithmetical progression, will here indicate an arithmetical
series of logarithmic values and hence a geometric series in
the original data — thereby instantly betraying to us the fact
that our phenomenon has for the length of the straight line
followed the law of organic growth. And the failure of our
curve to maintain a straight line will indicate the failure of
our phenomenon to follow the law of organic growth. All this
is as it should be, but the method of charting is tedious.
The other two methods open to us achieve precisely the
same resulting curve on the chart, but obviate the need of
turning our original figures into logarithms. No need to
bother with a table of logarithms, nor indeed, to understand
the so-called intricacies of such a table. The trick is turned by
RATE-OF-CHANGE SCALES
389
merely converting the scale of the chart, once and for all, be-
forehand, into a logarithmic scale. That is to say, we must
calibrate the scale figures for the natural numbers, but enter
these calibrations or scale figures at points on the scale which
are plotted, graduated, or measured, at the values not of the
natural numbers themselves, but of their logarithms. Such a
scale we shall throughout the remainder of this book call a
logarithmically projected scale.
PRICE OP POT'ATOES
Average Retail Price per Pound
United States
1913-1920
(Source:- Bureau of Labor Statiattos)
^^CJt-COlOt-tOCMCOfcO
<DlBp • m • * • • • •
Fig. 330. The Rate-of-change Curve— Second Method.
This scale carries the numbers, not the logarithms, but is not handy.
The first of these two simpler methods uses the same
arithmetically ruled co-ordinates which we have used for
amount-of-change curves.i The scale is therefore somewhat
unhandy. For if every equal interval or distance up the paper
or scale is to stand for an equal logarithmic value, it must
stand for an equal arithmetic or natural number ratio. If we
^ For a full description of this method, see Irving Fisher, llie Ratio Chart for
Plotting Sfati?ticsj American Statistical Association Quarterly, June, 1917, p. 578.
390
CHARTS AND GRAPHS
calibrate the first (i.e. lowest) abscissa (horizontal line) as
unity or 1.0, and let each distance or interval between the hori-
zontal lines stand for a 10% increase (that is ratio of y^), then
obviously we must calibrate the second horizontal as 1.1, the
third as 1.21 (that is ^ of LI), the fourth as 1.331 (that is
fj of 1.21), the fifth is 1.474, the sixth as 1.622, and so on.
This is what we would call an unhandy scale. It is difficult
to plot points on such a scale. Nevertheless, it can be done,
and the resulting curve will be the same as secured by the
previous method of plotting the logarithms of our series. And
we have avoided the task of turning each figure of the series
individually into a logarithm. And by either method you will
notice that we have been free to make our curve fluctuations
as high or as low as we wished, by merely selecting our scale
on a larger or smaller unit length.
The third method is, however, the best of all, for it makes
the plotting of logarithmic curves as simple and easy as the
plotting of arithmetical ones. It consists in using specially
ruled paper, provided by many publishers of chart paper, in
which the co-ordinates are unevenly spaced so as to correspond
with the logarithmic values of the round numbers in the
original series. Thus instead of an abscissa or ordinate at the
value of 1.21 (equidistant with the abscissa or ordinate of 1.0
from the abscissa or ordinate of 1.1), this paper has the ab-
scissa or ordinate of 1.2, slightly closer to that of 1.1. Like-
wise instead of an abscissa or ordinate for 1.331 (at another
equal distance), this paper has the abscissa or ordinate of 1.3
still closer to that of 1.2. So it goes throughout the scale.
The paper has been carefully ruled up with these gradually
diminishing distances or intervals between ordinates accu-
rately measured to correspond with the true logarithmic dis-
tances of the round numbers from 1 to 10 and all fractions
between these round numbers.
And since, as you have seen, the logarithms of every similar
succession of significant digits are the same (in mantissa), we
need merely multiply or divide these round numbers in the
printed scale of this chart-paper, by any power of ten to make
the scale suitable for our data. This is the same as saying that
we can shift the decimal point as far in either direction of
these printed scale figures as we please, and the paper will still
be properly ruled off and scaled. Again it is the same as saying,
that we may add or prefix as many ciphers as we want to these
RATE-OF-CHANGE SCALES
391
printed scale figures. The changing of the printed scale-figures
running from 1 to 10 into a scale in which the round figures
PRICE OP POEATCES
Average Retail Price per Pound
United States
1913-1920
(Source!- Bureau of Labor Statistics)
u u
O <D
ft's
U g
© 01 p
Fig. 331. The Rate-of-change Curve — Third Method.
The simplified and handy scale of original numbers.
fit our data is very easy. The only thing to remember is that
it is done by multiplying or dividing the printed scale figures
by a constant — ^whatever constant we please. In this it differs
from the changing of scales on the amount-of-change curves —
in which we could have used addition and subtraction. The
writing in of ciphers behind or before the printed scale-figures,
is merely a form of multiplication or division, in which the
constant is some power of ten. It is indeed perfectly possible
to use for our constant some figure which is not a power of
ten. But we must multiply or divide by this constant — ^we
cannot add or subtract.
Now it often happens that a scale running from 1 to 10,
that is in which the maximum of the scale is ten times the
minimum, does not afford us sufficient range for the fluctua-
tions of our data. Were we to attempt to plot our curve upon
this specially prepared, paper, we would find that the curve
would quickly run off the chart. To this problem the answer
is very simple. We merely ’join together two sheets of this
RJTE^OF-^CHJNGE SCALES
393
paper, or two sets of these rulings, and recalibrate the upper
one by adding an extra cipher to its scale figures. In this case,
the entire scale, through two sets of rulings, runs from 1 to
100 — that is, the first runs from 1 to 10 and the second runs
from 10 to 100. Simple, isn’t ill Not only two, but many
of these sets of rulings, can be joined together in this way,
giving us a scale range of from 1 to 1000, 10,000, 100,000,
1,000,000, or more. In fact, the publishers of chart paper
have anticipated this need, and provide paper with these sets of
rulings combined two, three, four and sometimes more, upon
a single sheet. Each set of rulings is called a ‘‘deck.” The
single-deck paper runs from 1 to 10, the double-deck from 1
to 100, and so on, two and three-deck papers being the most
generally useful ones.
These considerations of the number of decks needed for a
chart are all based upon the range of fluctuation in the series
to be plotted. Before determining upon the number of decks
to use in your chart, you must first glance through your series
and note not merely its highest but also its lowest items. If
both have the same number of (integral) digits, a single deck
is sufficient; but if the maximum has more (integral) digits
than the minimum, you need as many additional decks as there
are additional digits in the maximum figure. A little thought
will enable you to determine exactly the number of decks you
need before you start your chart.
A very different problem is the size of decks used in your
chart. In order to make their chart-forms of uniform over-all
size, the publishers of this paper are accustomed to make the
decks smaller as they join more of them together. Thus if in
single-deck paper the chart measures six inches to the deck, in
double-deck paper the deck will measure three inches that the
chart may still measure six, and triple-deck paper will have
three two-inch decks and in four-deck paper the four decks will
each measure one and one-half inches. This, for general ap-
pearance, is excellent, but you must not mix the various sizes
of decks in the same report or set of charts. You must main-
tain a uniform size of deck, for a chart upon a three-inch deck
cannot be compared with one upon a two-inch deck. The deck
on the smaller scale will show the same fluctuations smaller
than they would be upon a deck on a larger scale. So if you
use, let us say, the two deck paper in which each deck meas-
ures three inches, you must keep on using that paper so long
394
CHARTS AND GRAPHS
as you are charting statistics to be compared with it, regard-
less of whether your data at times calls for only one deck or
for three or more decks (in this latter case you must make up
three or more deck paper to fit the two deck paper, the decks
being uniform and the chart larger). So it is best, therefore,
400
375
360
325
300
275
250
225
200
175
150
125
100
80
Fig. 333. A Rate-of-Change Part-Deck Form.
This form, designed by the author, for price-fluctuations, is used by the Bureau
of Labor Statistics both as an office-form and in its publications on prices. The
horizontal faint-rulings (those without scale-numbers) are blue in the original,
so as to disappear in reduced reproductions.
before preparing a series of charts, to inspect all your data and
pick once for all the best paper suitable for the most widely
fluctuating series in the data.
In addition to the one, two, and three or more deck papers,
you may occasionally need, and can also obtain from some pub-
lishers part-deck paper. This is useful for showing fluctua-
tions which do not cover a range of more than 1 to 3 or 4, that
is, in which the maximum item is not more than 200 or 300 per
cent greater than the minimum item. Here you may find that
a whole deck would waste paper and fail to show the fluctua-
tions clearly enough. You may therefore feel called upon to
adopt a chart-ruling covering only a part of a deck. But, as
1 9861
RJTE-OF-^CHJNGE SCALES
395
will be later shown, it does not pay ordinarily to carry this
detail too far, for as you take a snntaller and smaller part of the
deck you will find the rulings approaching nearer and nearer
to plain arithmetical or uniform distances and your curve re-
sembles more and more closely a plain amount-of-change curve.
tfSTALS AND KETAL PRODUCTS' Copper, Ingot, Eloetrolytlo, N««r Yoric, Uonthly, ISIS-ISIS
(Before 1907, Lake)
owoo^iooiutr. o» w o
price rioc»So»o>oiSgooo>5o>o><»oioo®r“e-e“P“«owo»o>o^2
« <e oj o w
Fig. 334. Part-Deck Rate-of-Change Paper.
The curve-chart record form used in the Bureau of Labor Statistics for price
relatives. This is an office form— in this case filled out for electrolytic ingot
copper prices at New York — the chart nearly filling 8| x 11 inch paper sheets.
There is, however, still another form of ruling which, you
will often find useful, known as the ^^split-deck.’"' By means
of a split-deck you can often keep to larger decks though
your maximum and minimum figures in a series lie in addi-
tional decks. The split-deck is merely the upper half of one
deck and the lower half of another joined together in a single
chart. You may have split one-deck paper or split two-deck
RATE-OF-CHANGE SCALES 391
papers or more, but the first is the only kind ordinarily pub-
lished. By means of a split-deck chart-field you can often
keep to a larger size of deck and yet show a curve whose
maxima and minima lie in different decks. Any paper can be
converted into split-deck paper merely by multiplying the
printed scale figures by some constant other than ten or a
power of ten — ^two and five being the usual and most con-
venient constants for this purpose.
If you do not have access to the marketed forms of spe-
cially ruled logarithmic chart-paper, you can readily prepare
it for yourself, either by plotting for your scales along the
axes, the logarithms of the round numbers, as found by con-
sulting a table of logarithms, or by copying the calibrations and
graduations from an ordinary slide-rule. If the slide-rule (in
which the deck is usually about five or ten inches long) does
not have the right size of deck to suit you, and in general if
you wish to alter a scale as to size of decks, you can accomplish
this by the trick of laying off (or ^^projecting^’) the given cali-
brations to precisely the size you desire by the use of parallel
lines from the given scale to the desired scale, across a triangle
formed by the two scales and the last parallel.^
It is a pretty way of ornamenting the page on which a
rate-of-change chart is shown, to mark off a short additional
scale near, but not as part of, the chart. This scale need not
be as long as the scale upon the chart. It may be made up of
two parallel lines close together, with cross-lines at the round
numbers on the scale, the whole looking very like a long narrow
ladder. If you wish to make it more conspicuous, the alter-
nate spaces between cross-lines can be blacked in. The virtue
of this gratuitous and emphasized scale is that it calls the
layman^s attention to the strange and unusual (to him) nature
of the scale of the chart. And its strong markings also show
the constant significance of distances upon the chart, regard-
less of height. By recalibrating this extra scale with scale-
numbers of ‘^per cent increase or decrease’^ (that is, 0 at the
point of 100, SO at the point of 150,-25 at the point of 75) you
make this constant significance of distances even clearer. The
small extra scale is then an excellent device for use with a pair
of dividers — it is like a scale of miles on a map. The reader
2Cf. Figs. 166 and 167 on p. 185.
Ibstraot)
CHARTS AND GRAPHS
t
I
r
I
I
O in' o p;:
._ii,, I t I I n n
RATE-OF-CHANGE SCALES
399
can adjust his dividers for any two points, hold the dividers
to the scale, and read the percentage relation between them.
And this, after all, is the purpose of rate-of-change charts.
Note to Fig. 336
Fig. 336 illustrates the computing possibilities of rate-of-change paper. The
data, as will be seen above the chart, is for periods of irregular length as shown
by the dates below. While the points have been plotted for the data, these points
cannot all be connected as a single curve. It is necessary, as pointed out in the
chapter on frequency curves, to find avemges for periods of uniform length.
We need not, however, calculate these averages — on the contrary with a pair of
calipers or dividers we lay off the proper distances below the points from the special
scale at the right (or, in its absence, from the regular scale at the left) and obtain
at once the plotting points for a single connected curve. In this chart two such
calculated curves have been drawn — one for an annual average and the other,
just one deck higher, for a decennial average,
RATE-OF-CHANGE SCALES
401
Fig. 337. One Way to Find the Rate-of-change Scale.
On any horizontal distance OX lay off OA equal to one-tenth of OX and project
a semi-circle on OX (see upper left-hand diagram); drop a perpendicular Ab from
OX to intersect the semi-circle at b. Then OA : Ob : : Ob : OX and = {OA)
{OX) — (1) (10) and Ob — 'V 10. Lay off OB on OX equal to Ob (see upper
right, hand diagram) and drop a perpendicular Be from OX to intersect the semi-
cir cle at c. Then OB : Oc : : Oc : OX and (9c2== {OB) {OX) = 10 VTO and Oc=:
\/l0\/l0. Likewise (see third diagram) lay off OC, OD, OE, and OF on
OX, These distances (see fourt h diagram) represent 10 \/ 10, V 10 a/ 10 a/ 10,
VloVloVloVlO and V loV loV loV lOVlO respectively, or 10/^,
\ 10 \ 10 V lOvlO and \ 10 \ 10 \ 10 v lOvlO respectively, or 10/^,
lOK, lO^^’^ifi, and 10®^^ while we already have in OA, OB, and OX, obviously 10®,
101^, and lOL Similarly (see fifth and sixth diagrams) we can find 101^, and
10^, lO^iB, 10^'-'^2 from semicircles upon OB and OC. The ordinates from these
points (see lower left-hand diagram) intersect abscissae from a scale of the expo-
nents, 0, ]4:, so as to form a logarithmic curve, or curve of the loga-
rithms of numbers, in which y = log;r. By taking abscissae from the intersect
points of this curve with ordinates from the even numbers from 0 — 10, we find
the logarithmic scale of these numbers (see lower right-hand diagram).
Chapter XXXV
RATE-OF-CHANGE CURVES
From the construction of the logarithmic chart, we turn
naturally to the general principles of its application, and here
we must consider two of its limitations. The first of these has
already been touched upon, that the utility of the logarithmic
projection is limited to data in which the range of fluctuation
is fairly great. There is not much to be gained from the logar-
ithmic projection when the maximum of a series does not vary
by more than a 100% from the minimum of the series. For the
logarithmic scale and the arithmetic scale approach each other
more and more closely as the percentage between the limits is
made smaller and smaller. Within a range of 100%, that is,
when the maximum or upper limit is not more than twice as
great as the minimum or lower range, the difference between
the two projections or scales is not great enough to justify
ordinarily the effort of the less usual method. Within a range
of 50% variation the approximation of the two projections or
scales is very close indeed, and within a range of 25% varia-
tion, the difference is hardly noticeable. The real value of
the logarithmic projection is for data which fluctuates to ranges
exceeding two or three hundred per cent or more.
This fact, that when only slight changes take place in the
total series under observation the geometric progression closely
approximates the arithmetical one, enables us to use the
latter, because of its simplicity, even for the purposes of inter-
polation and extrapolation. Thus while the population of the
United States increases by about two per cent per annum, a
geometrical progression, yet the Census Bureau employs con-
stant differences or amounts of change in estimating the local
population for the inter-censal years. The results are sufii-
ciently reliable, because at two per cent it would take the geo-
metrical progression many years to outdistance the arith-
metical progression noticeably. And in the charts which fol-
402
RATE-OF-CHANGE CURVES 403
low, designed to show geometric progressions instead of arith-
metical ones, it must be remembered that no great benefit is
secured from charts in which the range of fluctuation of curves
is slight.
ANNUAL
RATE
! I I I ^ L_J 1 1 ^ I L_1
1919 1920 1921 1922
Fig. 338. Comparison of Series Lying in Different Parts of Chart, Though
Not Fluctuating Greatly.
Annual rate of turnover of bank deposits in representative groups of banks in
different cities . — Permission of Air. Carl Snyder.
To this limitation of the usefulness of the logarithmic
chart method, we must note an important exception, arising
when a number of different series are to be compared and al-
though their individual fluctuations are slight, yet they would
lie upon far different portions of the chart. In the chapter
on Index Numbers, you saw that such quite different series
could be easily compared by reducing them to percentages, the
change into percentages or relatives making their fluctuations
comparable when charted upon arithmetical or ordinary
amount-of-change chart paper. The use of the logarithmically-
projected chart scale, that is, the rate-of-change paper, will
however obviate the need for mathematical computing re-
quired to change the series into relative percentages, as the
logarithmic projection makes their fluctuations comparable
regardless of their position upon the field of the chart. Thus
404
CHARTS AND GRAPHS
pfflM /iim PACioay toes
Average farm labor wagee (irher© board was not included) in the United State©
ccnipared with average weekly eaminge in repreeeritetive factories, H.Y* state
1910-20
(Souroesj- U* S# Dept* of Agriculture and K* Y* State Dept* of Labor)
Farm Labor g
by the month
without board ^
CO
to
Factory workers
(office and shop)
weekly earninge
CO
to to e* to
CO to *o
* • • •
CM <•«» to O
•H rHI CM
O
lO
two series of data which fluctuate similarly as to rate of change,
appear very unlike when plotted arithmetically, if one lies
further from the zero or base line than the other (as when the
units of measurement of one are millions of dollars, and of the
3.69 28.15 64.95
RATE-OF-CHANGE CURVES
405
FARM AND FACTORY WAGES
Average fam lebor wages (where board was not included) in the tJnited States
compared whth average weekly earnings in representative factories, N. Y* state
1910-20
(Sources:* 0# S* Dept* of Agriculx.ure and N* Y* State Dept* of Labor)
Farm labor
by the month
without board
Factory workero
(office and shop)
weekly earnings
o
IT
^ o>
«> to wo
1:8
W fH
U3
• P'3
o «H
OC* to
• <o
CO H
lO to
• U)
O H
U3 CO
• CO
to
o
• <VJ
CO CN|
Farm labor by the
day (not harvest) ^ <»
without board
to CM
• M
r-t
;0
• 00
CM rH
r-t ID
to CM
uaco
to CM
250
200
160
100
60
10
0
/
/
.
/
^
mm
mm
mm
■H
mm
mm
mm
mm
HHI
HH
mm
mm
mm
/A
mmn
mm
^m|
mm
mm
mm
mm
mmi
mm
■■ll
mm
mull
IM
gmi
mm
umi
1
1
imm
imm
mm
V'
mmgi
WSM
Qum
■■I
HH
BHI
HH
■ml
.
to t> ® <ft O
Fig. 340. Amount-of-change Curve (Relative Numbers).
Other dollars and cents). But when logarithmic paper is used,
these two series appear to fluctuate together (just as they did
when reduced to index numbers or relatives). The logarithmic
chart method moreover avoids all confusion which might arise
as to the base year or period employed for the two relative
series. This advantage becomes important when diflPerent
4o6
CHARTS AND GRAPHS
PAHM AUD PACTORTf WAORS
Average Perm labor wages (where board was net included) in the United States
compared with average weekly earnings in representative B*y. state factories#
1910-20.
(Sources:- U. S. Dept, of Agriculture and H* Y* State Dept, of Labor reports)
Farm labor o
by the month
without board
<0
lO
lO
f-4
6
to
Factory workers
(office and shop)
weekly earnings
to
lO «c
<0 o
r-l CM
periods of time are used as the bases for the two relative series,
for even identical series would differ from each other with dif-
ferent base figures when plotted upon amount-of-change paper.
The other limitation of the logarithmic or rate-of-change
chart (and a much more important limitation) is that this
RATE-OF-CHANGE CURVES 4°?
chart can be used only for values in which zero is an absolute
limit. The chart cannot be used for data in which the values
cross from positive into negative numbers or vice versa. Zero
is an absolute limit to any geometric progression and to any-
thing operating under the law of organic growth. The popu-
lation of a community can never be zero (if it is to remain a
population) and it cannot be a negative quantity. The pro-
duction of a factory might be zero, but it cannot be a nega-
tive quantity. The sales of a concern cannot be negative.
Innumerable examples could be given of data to which zero
is an absolute limit and to all such data the rate-of-change
chart is not only applicable but proper.
To the rule that zero values cannot be shown upon a logar-
ithmic chart, there is an apparent exception, to be found in
the case of data measured in units which are made upon an
arbitrary and not upon an actual, zero-point. The Fahrenheit
scale of temperature, for example, places its ^^zero^’ value a
short distance below the freezing point for water at sea level.
Zero here is an entirely arbitrary valuation and does not mean
an actual nothing, that is, it does not mean zero heat. To plot
temperature by taking the logarithm of Fahrenheit degrees
themselves would be ridiculous. Not only would we be unable
to show zero degrees Fahrenheit (because the logarithmic scale
cannot reach zero) but indeed the shape of any curve which we
might plot in this way would be meaningless. The sensible
procedure would be to plot the temperature after changing
the readings into the absolute scale of temperature, or more
simply, to prepare a special scale in which the degrees were
plotted at their values on the absolute scale. This is done by
graduating the scale according to the logarithms of the ab-
solute degrees of temperature, and then recalibrating or label-
ling these graduations with the equivalent Fahrenheit read-
ings. After this recalibration or special labelling the figure O
would of course appear upon the scale of the logarithmic chart,
having been entered at the scale-point which really represented
about 265, the point on the absolute scale corresponding to
Qo p This example makes clear that a zero reading or value
can appear upon a logarithmic chart when it is fictitious and
really represents a positive value. Another example of the
same type would be a scale of time which included the year 0
A. D., from which we date our years in modern history. All
such are cases in which the real values plotted upon the chart
4o8 charts and GRAPHS
have actual positive values, which through some peculiar
circumstances must be assigned zero or negative values to
conform to ordinary practice.
A more general example of this recalibration of the scale
resulting in zero and negative values is to be found in per-
centage scales in which the hundred per cent point or line has
been relabelled zero and all other figures on the scale corre-
spondingly relabelled to represent percentage of increase or
decrease from this particular point. What has really happened
here of course is that 100% has been subtracted from every
point along the scale in order to get the desired calibration.
It is an apparent (though not a real) exception to the rule
given in an earlier paragraph on the construction of logarithmic
charts in which scale changes were said to be made only by
multiplication or division of the scale figures, and not by ad-
dition or subtraction. Reading upward on this new ‘"^per-
centage increase or decrease’^ scale from “0’^ (entered at the
true point of 100), we find “+50"’ entered at the true point of
Femalea Kftloa
RATE-OF-CHANGE CURVES
409
ISO, “4-100” at the true point of 200, “4-900” entered at the
true point of 4-1000%, and so on. Reading downward we
find “ -10%” entered at the true point of 90%, “ -S0%”
entered at the true point of 50%, “ -90%” entered at the
true point of 10%, and so on, the chart approaching but never
reaching the point of -100%, which belongs to the real zero
value which cannot be shown upon the logarithmic projection.
Such a recalibration as this special “percentage increase or
ACCIDENT MORTALITY RATES
THortallly Rated per 1,000 Population of Specified Aecidentfi
United States
1910*1912
(Sottrcdf United States Bureau of Censue)
Dro]»min^ ooo^oooooooooooocjo
Fig. 343.
CHARTS AND GRAPHS
410
decrease^^ scale, though possible, is really little used and in
actual practice more or less exceptional. In common with
the examples given in the previous paragraph, it is always a
little puzzling because the uniformity of ^^decks'^ has appar-
ently been destroyed.
To the fact that the logarithmic charts cannot show a true
zero point, we owe one of its most important and unique fea-
tures, namely that the height of a curve upon this paper is
entirely without significance and the curve may be moved
MARRIAGE AND DIVORCE
Jfmber of marriages and divorces reported emd total population
United States
1887-1916
(Source:- Census Report)
Fig. 344. Several Scales in a Single Split-deck,
RATE-OF^CHANGE CURVES
4IT
bodily up or down upon the chart without altering the sig-
nificance of the curve-fluctuations. When we say that the
chart cannot show a zero point, we mean that you could con-
tinue the deck and ruling of the chart paper downward infin-
itely far without ever succeeding in reaching zero. You would
merely reach smaller and smaller fractions of positive values.
The true zero-point is located out in infinity. It is therefore
taking no liberties with the chart to slide one curve up or down
as far as you please to make it more easily comparable with
another curve, because you are not really changing the position
of the zero point (that still lies for both curves out in infinity).
If the two curves are upon separate sheets of paper we may
slide one piece further down than the other so as to bring the
curves into close association with each other. Likewise if we
plot both curves upon the same chart we can use a small scale
for the lower curve and a much larger scale for the higher
curve, and so superimpose one upon the other. (In this case
separate scale figures may be an advantage to the reader but
they are not essential.) This juxtaposition of curves is one of
the chief advantages of the rate-of-change paper and can easily
be carried so far as to make one curve cross or intersect the
other. It is however considered better practice to prevent the
crossing of two curves which have been brought together in
this way, by sliding one curve lower down and altering the
scale correspondingly. If you are intent upon a very clear ex-
position of the artificial nature of this juxtaposition, you can
wipe out a small portion of the co-ordinates of the paper
between the two curves so as to indicate a break or omitted
portion of the chart-field between them.
From this arises a very important use of rate-of-change
charts in the detection of correlation. For not only is the dis-
tance or interval between points upon the logarithmically
projected chart useful for comparing the successive items in
the same series of data, but it is also useful in comparing
corresponding points upon different series. The problem here
is not to scrutinize the slope of parts of one curve, but to
scrutinize the distance between two curves. This distance is,
as you remember, when a distinct parallelism or mirroring of
the two curves is noticeable, an evidence of that similarity of
behavior w^jich is called correlation. To some extent, corre-
lation can be discovered by the use of index numbers, or re-
lative percentages, which make the fluctuation in different
412
CHARTS AND GRAPHS
COLTURAL O&ORXa 07 tHS OSXtCD STaTSS
Howspupore and parlodleali publisbod, patanta loauad, atudents in eo liases uniTsrsitiss and
sohools ot teohnelogy, and voluaea in libraiYes of various eisas (over 300 volunas each befof#
1690 and over 1000 voluoas tharsaftor) conparad with the population in the United States*
1870-1920
(Souroa:- U. S. Statistical Abstraet)
s a
•0 *“
Feptilatlon jg
(July 1st) w
s' 2
Voivsttas in
publio*
soeial and
school
libraries
Patents
isBuad
jUen
Students ..
in eollagat
univarsitiaa
and schools
of technol-
ogy, total
Tonen
Fig. 345. Shifting Curves to Avoid Insignificant Crossings.
sets of data more comparable. But a far more precise com-
parison is afforded by the rate-of-change curve.* To obtain
an exact measure of the degree of correlation between various
data, it is of course necessary to fall back upon the mathe-
RATE^OF^CHANGE CURVES
413
matical operations which yield a ^^correlation coefficient/’ But
for* most purposes the graphic method afforded by rate-off
change charts is sufficient and it is always, of course, much
easier and more rapid.
When a great number of series are plotted upon rate-of-
change curves they can be compared in short order merely by
inspection. If they have been plotted upon paper which is
sufficiently translucent, closer inspection can be made by
the use of ‘^light analysis,” that is by laying one chart over
the other, holding the two of them up to the light, and sliding
one back and forth and up and down until it most nearly
coincides with the other. Mirroring can be detected by turning
one of the charts upside down and bringing them together for
light analysis. The great advantage of the use of the rate-off
change chart for correlation detection is due to the fact that
various ^‘lags” in the correlation of the fluctuation can be im-
mediately corrected by this method, whereas by the long^
mathematical process, a slight lag might be sufficient to wipe
out and conceal any correlation which might exist, even
though that correlation be most complete^ Even when the
mathematical processes are to be used, in order to measure
the correlation exactly, it is best to use the graphic method
first, in order that the mathematical work may be performed
only upon the series showing appreciable correlation and in
order that any lag which may be present can be corrected for.
In short, for the work of correlation studies, so essential to
forecasting, the rate-offchange curve chart is becoming recog-
nized as necessary. Extrapolation has already been discussed
as a means of forecasting or predicting the nature of future
developments and this paper will be found admirably adapted
to such work, being statistically indicated wherever the fluc-
tuations of data are logarithmically more regular than they
are arithmetically.
A word may be said as to terminology. The curve charts
which we have previously examined under the general name
of ''amount-offchange charts” are sometimes called increment
charts or difference charts from the fact that the fluctuation
of the curve plotted upon them represents increments added to
or differences subtracted from their previous values. They
are also sometimes called arithmetical charts, from the fact
that the straight line upon them represents an arithmetical
series. The charts to which we are coming and to which we
414
CHARTS AND GRAPHS
POP
MlH.
)00
«0
Qa
M
YEAR&
ACTUAL POPULATION OF THE UNITED STATES.^ DIFFERENCE METHOD.
Showing the impossibility of correctly comparing rates of increase at different periods.
From Irving Fisher's "'The Ratio Chart” in American Statistical Association Quarterly, June, 1917.
Fig. 346. An Amount-of-change Chart-
give the general name of ^Vate-of~change’’ charts are some-
times called ratio charts^ from the fact that the fluctuation of
the curve plotted upon them represents in a certain sense the
ratios of change rather than the amounts of change. The
name is a poor one because the ordinary bar-chart and the
ordinary amount-6f-change curve both express more graph-
ically the actual ratios between the quantities; the rate-of-
change chart showing graphically only the changes of ratios
but not the actual ratios themselves. These charts are most
frequently called logarithmic charts from the fact that they
show logarithms and not the anti-logarithms or natural num-
bers.
The real distinction between the amount-of-change curve
and the rate-of-change curve is the distinction between quan-
titative and qualitative analysis of data. For a quantitative
analysis, that is a study of the actual quantities involved, the
amount-of-change paper is necessary. But for a qualitative
analysis of the figures, that is a study of their comparative re-
lations, ratios, and proportions, the rate-of-change paper is
necessary. The fundamental distinction to keep in mind is
that this qualitative rate-of-change paper illustrates relative
or proportional changes and is significant only as to them.
Moreover, because this paper does not illustrate totals, it is
always well to have at least your important data plotted both
'apon amount-of-change curves and rate-of-change curves, that
from each type of curve you may easily get its particular sig-
^ The name, it is believed, was introduced by Professor Irving Fisher, ohiu cit
RATE-OF-CHANGE CURVES
415
VEAR&
. THE SAME. RATIO METHOD.
Showing clearly the slight deviations, since 1860 , from a uniform rate of growth.
F'rom Irving Fisher's “The Ratio Chart," in American Statistical Association Quarterly, Ju7ie, 1917 ,
Fig. 347. A Rate-of"change Chart.
nificance. And It is because the qualitative study, that is the
study of relative values, is so frequently the more useful one
that the rate-of-change charts are themselves so commonly
more valuable.^
2 The significance of the logarithmic projection need not be difficult to understand’
The measurement of star-light in magnitudes, and the measurement of sound-waves
by means of the octave and its parts, are familiar examples in which we have adopted
logarithmic (or exponential) units of measurement, as clearly necessitated by the
type of the phenomena. It is permissible to think, therefore, that in all cases where
the logarithmic projection is found suitable, nature is operating in logarithmic units,
that is, changing organically in geometric progression, while man is still thinking in
terms of arithmetical units, and must needs project these logarithmically,
Chapter XXXVI
HISTORICAL RATE-OF-CHANGE CURVES
For the curves of historical data we find a type of rate-
of-change chart which is fast increasing in popularity and
general use. Though it involves the logarithmic projection of
scale, yet that fact is so completely camouflaged by the rulings
of the chart (in ‘^^decks"’ as described in the last chapter) that
we need no longer apologize for its use in a popular publication
nor attempt to explain it in conferences. We merely murmur
something about its being a truer picture of fluctuations and
let it go at that. If the other chap does not understand — and
this includes chief executives and officials — he at least realizes
that he ought to understand and enters no protest. In short,
this chart form has already reached the stage of notoriety in
which it need no longer skulk about in laboratory corners, but
can parade in public with a slightly exclusive, but very ef-
fective manner.
The peculiarity of this chart is that it has a logarithmically
projected scale along one axis only, the vertical or y-axis.
Its x-axis is innocent as a new born babe of any such develop-
ment — that is to say, its A;-axis is projected arithmetically.
And for this reason the chart is often, in charting office par-
lance, somewhat crudely but tersely called ^^semi-logarithmic.’’
Technically, of course, it can only be described as “A:-arith-
metic, y-logarithmic.” That the form is appropriate, however,
for historical data, may be seen if we recall the law of organic
growth in which it was stated that growth by uniform per-
centages took place at uniform intervals or periods of time.
Indeed, as the law of organic growth essentially deals with his-
torical data, that is, data during various points or periods of
time, we may consider that chapter a discussion of the general
theory of the historical rate-of-change chart in particular.
The student may be surprised that time, in itself a natural
phenomenon, should defy the law of organic growth. Eminent
416
HISTORICAL RATE-OF-CHANGE CURVES 417
THE WOULD* S PHODOCTIOH
Esiiam.'iftd gold« iron« coal and ootton produotlon and population
World. Spoolfiod yeara, 1800-1919
(Hote;- Qold figurea ar« annual aToragea for erirrent dooados befer® 1900}
(Source:«> 0. S. Statistloal Ibttraet)
Fig. 348. Long-Time Series of Economic Data.
engineers "with exceptional mathematical ability, have ques-
tioned the logic of charting time arithmetically with logar-
ithmically projected dependent variables. The answer may be
found in the very close approximation of the logarithmic series
to the arithmetical one through small ranges. In the whole
history of time the origin is so infinitely far removed and the
range of known history so extremely minute (comparatively
speaking) that there would be no appreciable difference in the
resulting chart were the A:-scale plotted by either method.
While it is certain that for modern times the two would coin-
cide, it is furthermore impossible to fix the true origin of time.
For zoological and palaeontological charts the birth of the
moon, or some similar event, might be a useful zero-point, but
in business and in historical charts in general the result would
still be the same. In other words, we may consider the his-
4i8
CHARTS AND GRAPHS
VIOLENT-DEATH RAIjBS
United States, 1900*1920
<Scmrcee;- For Hoiaxcides, euioldes, total accidents and automobiles, the Census Reports;
For tynohings, the Tuskeoge© Institute; for street accidents. Dept of Health, N« Y* City. |
(Note;.? Figures in parenthesis are annuel arerages for 1901-1905.)
street accidents
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Fig. 349. Short-Time Series of Economic Data.
torical rate-of-change chart to be really logarithmically pro-
jected on both axes, with the :v-axis scale range, however, so
short that it appears to be arithmetical.
In the construction of historical rate-of-change charts, the
same principles apply (with the exception of the logarithmic
scale on the y-axis) as in the construction of historical amount-
of-change charts, to which several chapters have been devoted.
A standardized form is useful when many charts are being
made, most of the marketed forms described in the last chap-
ter being useful for this purpose, as they contain no ordinates
or vertical rulings and can easily be provided with the latter
HISTORICAL RATE-OF-CHANGE CURFES 419
to suit each case. The field should not occupy more than
half of the page (on an 8| by 11 basis) in order that data can
be entered in full. The same position of the field in the lower
right hand corner should be maintained, when the charts fit
together to form a series, in order that they may be closely
overlapped either horizontally (to show one continuous curve)
or vertically (to show seasonal and cyclic fluctuations). The
same problems as to plotting points of data at unequal time
intervals arise and they must be settled in the same way. In
general the historical rate-of-change chart is but a duplicate
of the historical amount-of-change chart, modified in regard
to its y-axis scale and the plotting of its dependent variable
data.
Not all the historical amount-of-change charts, however,
can be duplicated in this way. With the simple curve, there
is of course no difficulty (either in its usual or silhouette form).
But in the Zee-chart the cumulative is not profitably copied;
indeed it is rarely of any use to plot historical cumulatives (at
least when these are for limited and repetitive c^^cles or periods
of time) logarithmically. The bar-charts (except silhouette
bar forms of curves) should never be plotted logarithmically,
as by their nature they suggest and imply a significance in their
heights, and the logarithmic chart, as you know, is without
significance in the height of its curves. ^
^ It is here perhaps best to warn the student against too confident use of other
chart methods for the purpose of showing rate-of-change fluctuations. In previous
chapters the series of chain-relatives or chain-percentages has been discussed and it
has been pointed out that the chain relatives are the anti-logarithms of the logarithmic
differences between successive items m a historical series. As the rate-of-change
chart shows by the fluctuations of its curve the amount of these logarithmic differences,
it is easy to see that the fluctuations of a rate-of-change curve do not represent the
chain-relatives or chain-percentages, but represent the logarithms of these chain-
percentages. Attempts have therefore sometimes been made to present graphically
the chain-relative series itself. A little experimentation however will show that the
method is not for most purposes useful. To be strictly accurate, the chain-relative
series must not be shown by a single continuous curve but by a number of successive
curves in which each plotted point is connected with the previous ordinate at its
intersection with the 100% line. l‘he result is a wholly disconnected picture and is to
some extent liable to wholly meaningless- changes of form when the time units of the
data are shifted or changed. Nor is it possible to present a more connected story
by joining each of these disconnected curves and plotting these new points the proper
distance above or below the last preceding points instead of above or below the 100%
line, for this would amount to a cumulation of the chain-relative series, additively,
when the series properly cumulates only by multiplication, and it w-ould result in a
curve in which no uniform scale is possible for the dependent variable, and in which
points at the same height upon the paper have different meanings or values at different
portions of the series. Except in very special circumstances the rate-of-change curve
chart is the proper method for showing the rate-of-change of fluctuation in data.
Fig, 350, A Careful Positioning of the Curve.
Showing retail price of all articles of food, combined, in the United States.
It would have been even better to begin the chart with 50 for the base line, as the 200-line would then be just twice the height of the 100-line
and the similarity to an amount-of-change curve would have been more complete . — From the Bureau of Labor Statistics "Monthly Labor Review F
HISTORICAL RATE-OF-CHANGE CURVES 421
Where the charts are to be used popularly it is well to go
to considerable pains to lessen the misunderstanding of the
chart by people who do not understand its logarithmic nature.
Some misunderstanding is bound to arise, but one of the sim-
plest and most effective measures is to position the curve (where
one only is shown) at about the height upon the paper at which
an amount-of-change curve with similar fluctuations, would
appear. Thus if your curve covers a range on the y-axis of let
us say from the 100% point to the 200% point, the highest
point on the curve is obviously double the absolute value of
the lowest. Now if we cut off our chart-field at the 50%
point, then clearly we will have equal distances between the
bottom of the chart, the low point on the curve, and the high
points on the curve. Now when Mr. Average Man — and it
may be Mr. Average Congressman and seem very important
to you — picks up this chart, he goes through the following
motions: “Ah, one of those damned curves — ^what the deuce
does it mean? Oh, profits on rotten meat. Well, I see they
doubled during the war. I can understand this thing easily.”
And then he drops it again. He does not notice that the bot-
tom of the chart registered 50% and not zero on its vertical
scale, but that does him no harm. He has at least seen one
thing truly, the relation between high and low points.
In business reports it is always a good plan to present both
rate-of-change and amount-of-change curves simultaneously
for all important historical data. The two charts of the same
information should be face to face so that the reader is con-
fronted with both at once, and cannot mistake the rate-of-
change form or judge quantities by its curve. Nor is this
useful merely to avoid mistakes on the part of readers uncon-
versant with rate-of-change paper. There is often a real use
for quantitative analysis of the data, which can be seen only
from the amount-of-change paper. The rate-of-change chart
is indeed for most purposes more effective, but it fails wholly
to give any picture of total sizes or quantities involved, and
is not the panacea which some of its enthusiasts would have
us believe. The whole truth about a series of data requires
not only the rate-of-change but also the amount-of-change
method.
In more scientific reports and in records which will only
be used by those who cannot misunderstand them it is often
useful to combine the two types of curve for the same data
422
CHARTS AND GRAPHS
B-RATiO chart
SHOWING RATE OF CHANGE
A-INCREMCNT CHART
Showing amount of change
Fig. 351. Comparison of Rate-of-Change and Amount-of-Change Curves.
The chart shows the growth of one dollar (lower curve) and six dollars (upper
curve) at compound interest . — Permission of Mr, John Wenzel.
upon a single chart, superimposing one upon the other. The
simplest method of doing this is to plot one curve for the
natural numbers and another curve upon the same chart, with
the same arithmetical rulings, for the logarithms of the data.
The first or natural number curve can be labelled Y and the
second or logarithmic curve can be labelled Log Y. If the
paper used for the chart is fairly translucent, it is not neces-
sary to look up the logarithms in plotting this second curve,
but a sheet with heavy logarithmic ruling can be placed
underneath the chart and the two sheets held up together to
the light while the logarithmic curve is plotted, from the
logarithmic ruling on the lower sheet, which are not intended
to appear upon the chart itself. The slide-rule will serve
equally well, if its scale is appropriate, an engineer’s scale
being used for the arithmetic curve and the slide-rule for the
plotting of the logarithmic curve. A clearer method of show-
HISTORICAL RATE-^OLSCHAAGE CURVES 423
ing the distinct nature of these two superimposed or combined
curves than by merely labelling them Y and Log F, is to trace
or draw in a small portion or zone of typical logarithmic ruling
immediately around the logarithmic curve, thus showing that
the latter has been cut out from a logarithmically projected
chart and inserted upon the arithmetical chart. Needless to
say, the combination of these two wholly different types of plot-
ting of a curve should be made upon a single chart only when
that chart will show a single series of data, for obviously several
curves brought together in this way would become confusing.
No discussion of the historical rate-of-change curve is com-
plete without mention of its value in forecasting, The use of
TltADE UNIOM MSaiBJjktbHIP OP THE WOkLD
l>iximbar of memborsi in ZO coxmtries
1910-1919
(Source:- Interimtioiml Labor Office^
Compare the extrapolated forecasts with those in Fig. 175 .
the logarithmic projection in general for correlation studies
has already been noted in the last chapter. But there is a
very special usefulness of the rate-of-change chart in predicting
from the course of past events, the probable course of future
events. The curve is merely extrapolated to the ordinate of
CHARTS AND GRAPHS
424
the desired point In the future and the intersection point is
read from this extrapolation. The degree of success or reli-
ability achieved in such methods of prediction depend of
course upon the faithfulness with which the future develop-
ments follow the course of the past record. But an estimate
of the probability of this can often be made from an inspection
of the past course of the curve itself. The method is only sta-
tistically indicated where the curve of the past shows very
regular and uniform trend. Through a combination of causes,
it often happens that these estimates can be more reliably
and usefully made by extrapolation of the rate-of-change curve
From Joseph E. Pogue's "Economics 0 / Petroleum"
Figr. 353. Careful Extrapolation.
A straight line (dotted) has been fitted to the curve and projected ahead ten years
to give forecasts. In 1930 the forecast is about 1250 million barrels as against
SOOmillions indicated by extrapolation on amount-of-change paper (see Fig. 176).
than by extrapolation of the amount-of-change curve, for not
only is the trend of the past often more uniform upon the for-
mer, but also in general it is true that the phenomena which
show uniform rate-of-change in the past can be more relied
upon to maintain their trend than the phenomena which show
uniform amount-of-change.
The historical rate-of-change curve is indeed one of the
most important instruments in the treatment of ordinary
HISTORICAL RATE-OF-CHANGE CURVES 425
business statistics and economic studies. It has not yet
reached the height of its vogue but it is fast gaining in prestige
and is fully deserving. You will not only find it illuminating
and fascinating to work with, but you will soon discover, if
you have not previously used it, that it opens up to you a
new world of investigation and research.
Chapter XXXVII
LOGARITHMIC FREQUENCY CURVES
Hamlet: Do you see yonder cloud that’s almost in shape of a camel ?
Polonius: By the mass, and ’tis like a camel, indeed.
Hamlet: Methinks it is like a weasel.
Polonius: It is backed like a weasel.
Hamlet: Or like a whale?
Polonius: Very like a whale.
This same doubt and debate arises over the shape of every
frequency curve. Such curves commonly present the widest
variety of shape and contour. It is indeed possible to con-
ceive of frequency data for almost any curve which may be
imagined. Some attempts have been made to classify these
various forms and divide them into a few typical groups.
These classifications rest upon the relative positions (along
the A;-axis) and magnitude (or amplitude) of the peaks and
valleys in the curve. They are restricted to simple curves,
presenting not more than one peak and two valleys (strictly
speaking, half-valleys) or one valley and two peaks (half-
peaks). Where more peaks or valleys occur in the curve, and
these cannot be smoothed out by applying a process similar
to the moving-average or total process, it is rather assumed
that the curve is not simple, but compound, being really a
combination of two or more separate simple curves. The
breaking down of a composite curve into simple curves is a
problem of advanced and difficult mathematical steps, super-
ficially not unlike harmonic analysis, and cannot be discussed
here. Outside of the engineering and scientific fields, these
curves will be rarely met and the student of business and eco-
nomic statistics need only acquaint himself with the treat-
ment of simple frequency curves. We shall first consider these
simple frequency curves as they appear on amount-of-chanee
plotting paper.
426
LOGARITHMIC FREQUENCY CURVES
427
While no classifications have proved very useful, the best
which has so far been formulated, and one of the simplest, is the
classification of Professor Yuled Yule finds four common types,
VITAi SUPERIORITY OF THE FBHALB
Percentage Exo«a« of Male ever Female Death Sat«
England and Wales
1851-60 and 1901-10
((Source:- Reports of Regiat-rar Oeneral of England and Walea)
Fig. 354. Compound Curves-
which he calls, respectively, the symmetrical curve, the moder-
ately asymmetrical curve, the extremely asymmetrical or /-shaped
curve, and the t/-shaped curve. These names are self-explana-
tory. In the first or symmetrical curve, there is a single peak,
from which on both sides the curve slopes away symmetrically.
The salient facts about this curve are its width or spread
(‘‘range’')? the height of its peak ordinate, and the relative
heights of its ordinates at regular or certain irregular (‘^quar-
tiles,” ‘Cecils”, or “percentiles”) intervals away from the peak
ordinate. The last detail would tell us whether the peak was
narrow, indicating great concentration of the observations
about that point, or wide, indicating a scattering or “disper-
^ Cf, Yule, Theory of Statistics.
428
CHARTS AND GRAPHS
sion^^ about the point, and there are mathematical methods of
expressing these details which may be found by consulting the
statistical authorities.
OUTPUT OP COAL MINERS
Average Ntimber of Tons of Bituminous Coal Mined by Pick-miners
per a-hour day in 118 mines, 17 states (all fields)
United States
1919
(Source:- Ethelbert Stewart)
Nximber Humber
of of
Tons Miners
Under 3 577
3- 4 913
4- 5 1251
5- 6 1491
€ - 7 1395
7- 8 1275
8- 9 984
9- 10 672
10 - 11 462
11 - 12 290
12 - 13 171
13 and over 336
Fig. 355. A Frequency Series Which Appears Slightly Asymmetrical.
The second, or moderately asymmetrical curve, presents the
same general form as the first, but its peak is no longer in the
middle of the curve, being nearer to one end than the other.
Consequently the curve does not fall away on both sides with
symmetry. Here we have a new element which is statistically
known as ‘^skewness’^ the curve being skewed over to one side,
and there are mathematical methods of describing more or less
explicitly the degree of this skewness.
In the third or extremely asymmetrical type of curve, we
generally have but half a peak and half a valley, that is, the
curve begins at a peak on one side and slopes down toward a
valley, but the other half of the peak, the opposite slope, as it
were, may be lacking. Where the opposite slope is present, it
is very near to a straight vertical line and the curve is merely
extremely asymmetrical. When it is lacking, the curve is
called /-shaped and not only the lower or valley end, but also
the peak end, of the curve may be “asymptote,’’ to the axis
of the chart. By an “asymptote” is meant a line which, while
LOGARITHMIC FREQUENCY CURVES 429
approaching infinitely near to an axis, gives no promise of
actually meeting it, no matter how far it be extended outward
along the axis.
THE DURATION OP MARRIAOES
OlTorcea Classified by Years of PreviouB Married Life
United States
1887-1905
(Source:* United States Statistical Abstract)
Number of
Divorces « ^ ^ <»“ uT
(Tota 5 900,684) « ® cva ^
10 CO C^il
•-4 Csj CM to lO •<*«
Years of Married Life
Fig, 356. An Asymmetrical Distribution.
The dotted line is the same curve plotted upon a logarithmic horizontal axi.s.
Note that, if this plot could be extended through the first open group, it would
probably become approximately symmetrical.
The fourth type, the [/-shaped curve, is one which describes
a letter U, being made up of a valley and two half-peaks. And
though it may be a very sharp or pointed, V-shaped valley, it
is perhaps more frequently found to be more or less flat, just
the reverse being generally true of the other three types. And
it might be added that this [/-shaped curve may be found in
both symmetrical and asymmetrical forms, according to the
presence or absence of skewness. Yule, however, makes no
subdivision of the U-shaped curves, as they are comparatively
rare.
It is now our purpose to show that these four more or less
distinct types are often interchangeable forms, which can be
evolved by different statistical and graphical processes, from
the same original series of data. For this purpose we naturally
regard the symmetrical type of curve as the more desirable
form, and shall consistently refer other forms back to it. The
reason for this, as has already been indicated, is that the more
430
CHARTS AND GRAPHS
regular and symmetrical a curve be, the safer and more trust-
worthy would seem its use for interpolation and generalization.
The symmetrical form is not significant only to the mathema-
SKY CLOUDINESS
Distribution of 3,653 Obeervod Intensitiei of Cloudiness
Breslau
1876-1886
(Source*- Yule, Theory of Statistics)
Fig. 357. Yule’s Example of a U-shaped Distribution*
tician who will seek a general equation for the phenomenon
described on the chart, but it is also more significant to the
layman, who, having once seen the symmetrical curve, will not
so easily forget it. And at this point we enter the subject of
the logarithmic projection of the scales.
To begin with the second, or moderately asymmetrical form
of frequency distribution, we have to note that its curve can
often be made symmetrical by plotting upon a logarithmic
scale. Plotted on rectilinear co-ordinates, that is, on the
amount-of-change plotting paper, the right-hand ‘^taih^ of the
curve, that is the slope toward the half-valley at the right-hand
side, is generally longer than the one at the left. At the same
time, the values of the independent variable, that is, the values
of the points along the A:-axis, are larger at the right-hand end
of the scale than at the left. So it is obvious that a logarithmic
projection will shorten this part of the scale in comparison with
the rest of the scale (for the log projection always condenses the
larger values). The result is to shorten the longer tail, often
LOGARITHMIC FREQUENCY CURVES 431
enough to produce absolute symmetry in the two slopes of the
curve.2 Nor does this process need to be wholly empirical, for
the student will soon learn to detect in advance the curves
COLLBGI. SALARIES
Sal&riee la Aaorican Collegos tnd UniTer«lii«a
Including Public and prirate Inatituticns
Onitad States
1920
(Source: U. S* Bureau of Education}
Ajtsisttints g
Inotruotors ^
Assistant Professor 6 ®
<0
Asscciato ProfesBora
Pull Profccsoro ^
Dollar* of Salary
Fig, 358. Six Moderately Asymmetrical Distributions.
Note that these have been made more symmetrical as to their sides and more
rounded as to their peaks by means of logarithmic scales.
“ See Fig. 361 on p. 435.
432
CHARTS AND GRAPHS
which;^ by this treatment, may become symmetrical.^ The suc-
cess of the method depends, of course, upon the nature of the
independent variable, whose values appear as calibrations or
scale-figures along the ^c-axis of the chart.
From the above it is clear that it would be impossible to
apply this method directly to data in which the independent
variable includes both negative and positive values and crosses
through zero. When zero is not a limit of the range, but is
included inside the range, it would be necessary, if the method is
to be applied, to change the variable to values which are wholly
positive or negative. This can easily be done, if the zero-
value proves on inspection to be purely nominal, arbitrary, or
relative, and to have no real zero meaning. Thus if the series
SENT INCEEAEKS
Number of Paailies Reporting Increases of Rent
(Jorerment Employees, Washington, P. C.
Year Ending Oct. 1, 1920
(Source: - Monthly labor Review),
0
1,983
•1 •
9
83
10 -
24
648
2S •
49
480
60 •
74
142
75 -
99
41
100 and over
29
Fig. 359. The Independent Variable is Measured from an
‘Arbitrary Zero Point.
be so arranged as to show deviation from its mid-point (median),
mode, mean, or from some other particular value, the devia-
tions, being measured above or below this point, will show in
the table as positive and negative stubs, and the point itself will
show as zero- By merely adding to all stubs the true value of
this point, we return the data to its primary form and wipe out
the false zero-value. In other cases the zero-value, though
arbitrary, cannot be so easily given its true value, and the work
is more difficult. When zero is a limit of the range of the data,
and is actually met in some of the data, the log-chart can be
used, but of course the values of the curve in the first interval
(next the zero point), cannot be plotted, as that interval becomes
» See Chapters XXVII and XXVIII,
LOGARITHMIC FREQUENCY CURVES 433
infinitely long,^ In the same way, the final class or group
cannot be plotted where it is indeterminate, for it too is
infinitelylong.
In general, the same considerations apply to the possibility
and usefulness of the logarithmic projection of the a:- axis scale
as are applied to the logarithmic projection of the y-axis scale,
already discussed in the foregoing chapters. The logarithmic
projection would seem appropriate whenever zero is an infinites-
imal limit (approached but never reached) to the independent
variable. Such a condition is inherent in the nature of the
phenomenon and can be detected immediately therefrom.
Most economic data are susceptible to the process of log
projection. It is noteworthy that each of Yule's examples of
the moderately asymmetrical curve can be plotted on logarith-
mic paper and made symmetrical thereby. Human beings, for
example, cannot have a negative height, nor a zero height;
communities cannot have zero populations; manufacturing
establishments cannot have zero employees; nor farms, zero
acreage. Frequency series of such phenomena, classified as to
their sizes, seem to call for logarithmic projection a priori. On
the other hand, profit or loss can be negative, as can net worth
and balances of all sorts, including ‘^stock’^ or ^^fund^^ data,
time and space dimensions with reference to particular points,
and for such data the log projection would ordinarily be both
impossible and meaningless.
Whether or not the vertical or y-axis be given a log projec-
tion is relatively unimportant. Ordinarily when the A?-axis is
so plotted, the y-axis can be also, and the resulting curve shows
a more rounded, less pointed peak. This may or may not be
desirable. For identification with the normal curve described
in the next chapter, perhaps more often the arithmetrical pro“
jection of the y-axis, even with the log is desirable, but it may
be that the identity will be established only when both y and
are logarithmic.
The third, that is, the extremely asymmetrical or /-shaped
curve, presents two possibilities. If it is extremely asymme-
trical, but has two ^'tails’^ or half-valleys about its peak, it may
belong in the same category as the second or moderately
asymmetrical curve. The difference in the degree of skew,
which seems so much more violent in the third than in the
^ See Fig. 3S6 on p. 429.
434
CHARTS AM) GRAPHS
LENGTH OP WORDS
Distribution of 10,000 Words by Number of Letters in Them
(Source:- BoTrloy, "Elements of Statistics")
Number
of Words
O O Oi H
CD <M
(M ’rH
Logarithmic
Number of Letters
Fig. 360. A Moderately Asymmetrical Distribution which the
Logarithmic Scale Has Not Made Entirely Symmetrical.
second, may be found wholly due to the greatly extended range.
Thus if we classify farms by acreage, our table may include
farms of less than three acres and farms of more than a thousand
acres- Here the range is very great, being through three
logarithmic decks, and the arithmetical projection has obviously
brought the log mid-point very, very close to its left end (and
LOGARITHMIC FREQUENCY CURVES 435
with it the peak, resulting in extreme asymmetry. The height
of human beings varies through no such great range and hence
shows, when arithmetically projected, only mild asymmetry.
SIZE or FAfiMS
0ait9<l St&tvs
1920
<8ouro» - C»asm*}
iTortig* lunbar ef SSS S 2 C
Farmt aacn ''.•’.''-'t •
6lnel#-<wire aub-greup ** S 2 S 2 2
In •nch Clnta
fci»l Bumbar of f«r»a
is a«oh ClsM
s
I
s
2
Fig. 361. An Extremely Asymmetrical Distribution Made Symmetrical
by the Logarithmic Projection.
Note that the left ‘'tail” of the lower curve has been extended through the open
class (zero to two acres) as a straight line, and therefore obscures the symmetry.
Large ranges are extremely common in business and economic
statistics, and the data will nearly always fly storm signals
indicating the need for logarithmic projection. Thus the usual
classification of cities by sizes runs through intervals with
£>i/.E or
Huaber of workers involved in Btriies
United States
191 «- 1921
(Somrce;- Monthly Labor Review)
Claeeee: Ko. of workers
Ko. of equiv. eub-olasees
\pprox, ntunber of workere
3001
1-10
1
11-25
1.5
26-60
2.6
197
(197)
346
(230)
412
(164)
164
(164)
296
(197)
341
(136)
143
(143)
268
(17S)
334
(133)
170
(170)
279
(186)
333
(133)
IBO
(150)
299
(199)
313
(125)
219
(219)
286
(191)
262
(101)
«6"
•’IS”
"se”
251-600
25
601-1000
60
over
348
(13.9)
238
(4.8)
254
284
(11.3)
193
(3.86)
286
278
(11.1)
141
(2.82)
216
339
(15.6)
206
(4.1)
576
262,
(10.6)
136
(2.72)
196
153
(6.13)
101
(2.02)
140
”376’'
Sgg8 8 888
H N ^ «0 00
iJ 1 ^- LJLLLLL L 1 1 L J
S S ^§§8 8 8 8
Fisf. 362. Th<* Doubl«-Logarithmic Projection is Best.
LOGARITHMIC FREQUENCY CURVES 437
various division points, such as 1000, 2500, 5000, 10,000,
25,000, 50,000, 100,000, 250,000, 500,000, 1,000,000. Who
cannot see at once that these intervals are merely round num-
bers approximating, as closely as convenient, equal geometric,
and not equal arithmetical, intervals?
When, however, the curve is /-shaped, that is, has only one
tail, another possibility creeps in. For it may be that what
we are treating as a /-shaped frequency distribution is really an
ogive, that is, a cumulated frequency series. The cumulation
of a distribution which is symmetrical on an arithmetical
projection is easily detected, of course, but the cumulation of
other forms may be neatly disguised in the description of the
series and pass for a time unnoticed. The behavior of these
other cumulatives we will discuss shortly, but it should be borne
in mind that the most frequent example of the /-shaped curve
is an ogive or curve of a cumulated series. And when you
meet a /-shaped curve, examine it first to make sure that it is
not an ogive. If it is not, then the possibility remains that
it is a very extreme asymmetrical curve, which is really not
/-shaped at all, and has two perfectly good tails, but one of
them is so very short as to be swallowed up in the peak.^ Thus
Yule’s illustration of the /-shaped curve, being the un cumu-
lated distribution of personal incomes in Great Britain, is as he
himself says, not really /-shaped, but merely so asymmetrical
that its lower portion has been swallowed up in the mode of
the series.
As to the fourth type, the i7-shaped curve, rare as it is, it
presents two or three obvious possibilities, one or another of
which may serve in its analysis. In the first place, we must
note that it may be merely an approximation to a distinct
‘‘yes or no” tabulation, its two terminal maxima representing
the two alternatives arid its intervening minimum the more
infrequent compromises. Thus a tabulation of eyes by degrees
of blue or brown color might show many wholly blue and many
wholly brown eyes with relatively few eyes of the various inter-
mediate shades, if the whole matter of color be a resultant of
the presence or absence of dominant color determinants. Yule’s
illustration of this type of curve suggests a similar condition,
being a record of sunshine and cloudiness at Breslau. If clouds
be a local evidence of a falling barometer over a wide area,
See Fig. 362, page 436, where this has happened in the last year.
43 B
CHARTS AND GRAPHS
and clear skies of a rising barometer, since it is obvious that the
air pressure can shift only one way or the other, we might be
justified in expecting the local conditions to show few inter-
mediate results. Closely allied to this is a second possibility,
namely that in a particular [/-shaped curve we really are not
dealing with a simple curve, but with a compound one in which
two opposite, extremely asymmetrical and apparently /-shaped
curves have been combined.
As a third possibility, we have to note that since a [/-shaped
curve is merely one of the first or second curves upside down, it
is possible that by taking the remainders or complements
(especially if the data be in p ercentage form) of the dependent
variable, we can right it again, taking the data out of the fourth
class entirely and throwing it into the first or second (or even
third) class, there to be treated as above outlined. And just
as the symmetry of the minimum-maximum-minimum curve
may be effected by the logarithmic projection, so the maximum-
minimum-maxiitium or the [/-shaped curve may be similarly
converted from asymmetry to symmetry. Mortality rates
often show an extremely asymmetrical [/-shape, which can be
changed to a valley of perfect symmetry by the log-scale, giving
a beautifully rounded curve with clearly emphasised variations
for different racial and occupational groups of the population.^’
And by subtracting the death rate in each age from the base
of the rate, we obviously get what might be called a survival
rate which, though less known, is a clear example of the second
group of curves.
It is not to be understood from the foregoing discussion of
statistical and graphical methods of producing symmetry in a
curve, that all frequency series can be made symmetrical by
proper treatment. Sometimes the very failure of the series to
be symmetrical is of prime importance and while we might by
round-about methods produce symmetry, yet the inappro-
priateness of these methods would be so great as to make
symmetry meaningless. Nor is it true that all frequency curves
will fall into one or another of the four classes mentioned. The
point of what has been said in the foregoing discussion is that
it is sometimes, indeed often, possible by very simple steps to
®The Census Bureau in its Life Tables has published elaborate charts of these
curves, but unfortunately through the use of arithmetically-projected scales, it has
been obliged to break up each curve into three or four parts, each part with a suitable
but different, scale.
LOGARITHMIC FREQUENCY CURVES
439
PBMAIB MORTALITY RATES
Uni tad States
1910
(In percerxtagas )
Sonrce:— U.S. Census, 19L4.
C- oo 0> O O) <0 CO 6 to o to
« rt
Teare of Age
Fig. 363. An Extremely Asymmetrical U-shaped Distribution Brought
to a Beautiful Symmetry by the Log-scales.
Note the emphasis given to significant ii regularities, such as the increased
mortality-rate at adolescence, among negroes, and its absence among the foreign-
born.
%
attain the desired symmetry, and discover an underlying
regularity in the behavior of the phenomena observed. To
determine whether the symmetry is desirable and significant,
and to interpret the meaning thereof when the symmetry has
been secured, is indeed a task calling for experienced judgment.
Furthermore, to detect the possibility of such symmetry
28.1
440
CHARTS AND GRAPHS
r, ^ . - mortality rates
oeeths at Each Aga.ohoan aa a Eareantaga of Thoaa Living at the Beginning of the Travloua Yean
The United States
1901 and 1910
(Source:- U« Sa Census Bureau)
<1^ ^
w to tOHOwr* o> to<«)<c.»aj w cjcuHcnc* lO'fonio
00 to tOOtOCD'f* H (OCOUOOJ aj* Wt>v-ICI>C- C~HHT0tI«
e- lO M<tOtOWW W WWTOal* U> tOt-C'OtO lOmCiaJ'liO
0> lOOrtuOt- <0 t»t“0>t0
OOHafO lO C-^TOIO
t> <£)U5a(J*tOCU Cl «tOa)<lO
C0C-t00>0> WaJ'OWt-l
C-C^C3tOO> tOafiOlOlO
t-coa*H<o locic-e-co
Cl lO I-I to w
Fig. 364. Same as Previous — Historical Comparison.
LOGARITHMIC FREQUENCY CURVES 441
requires imagination and familiarity with statistical data
which the student will not quickly achieve.
Some idea of the sign posts which indicate the appropriate-
ness of the logarithmic projection has already been given.
When the class or group limits which break the phenomena into
a series are at widely varying and rapidly increasing intervals,
such as (0), 1, 2, 5, 10, 20, SO, 100, 200, . . . (infinity) or 1
month, 2 months, 3 months, 6 months, 1 year, 2 years, 5, 10,
15, 20, 25, 30, etc, years, and similar arrays, then the geometric
CONVENIENT GEOMETRIC INTERVALS
Fig. 365. A Table of the Convenient, Nearly-geometric Intervals by
which the Range Between Successive Powers of Ten May be Divided.
nature of the progression is evident. But the law of organic
growth applies in far more cases than those which wear this
obvious marking. It may not be amiss to note some of Yule's
illustrations, since we have followed his classification of curves.
As he says, the symmetrical curve is rare in economic statistics
and his one example of it, relating to the stature (height) of
groups of adult men, covers so small a range in inches (from
58 to 76) that the logarithmic projection of the scale would not
appreciably alter it. Hence we may conclude that while this
particular series is practically symmetrical — on arithmetical
projection — yet it may really call for a logarithmic projection
because of the organic nature of human growth, and the arith-
metical symmetry may be entirely accidental. This idea grows
stronger as we observe that his illustrations of moderately
Number of Cases
442
AMERICAN ACCICENT TABLE
tluration of Temporary Total Dleabllity
(95,388 oaaee per 100,000 accidonts)
United States
1919
(Source:- Olive E. OutuRter)
Cases
IS
a>
OCMWttSOCUOie-UO^lOWCVjeUr^iHi-IM
a ♦>
tl
Period
period
essscscsssssssssstsrsss
O»- 4 M« 0 ^UJt 0 t-C 00 »O<^N»e'i|'«l
)eOrt*IO(Ot-OOOkr-li-<rHH>-4r-li-«i-l--li>4CMNMO}CMr
iittitillilii itiliiiit
cMK»^uj<or“COO»Oi-<e»»(o^u>«>t-«oa» 0 '^Mrt''
•o< 04 M^u 3 (o»«o-e'Ou>(aaoo
CM«lX>f«ll 0000 >to^t-C-< 00 >
C0O04OCMU> C0OOt-'«SW00i-»
COaOt-tCU}'<l«‘<titOlOCUt\>CM«-«M
>» >.
« AC CSC rsescssc
tj -o
Fig. 36 $. The J-shaped Distribution.
(Shown by the dotted line and plotted upon the outer or arithmetic scales)
becomes more rounded as logarithmic projection (shown by the full line and
plotted by the two inner or logarithmic scales) is used.
LOGARITHMIC FREl^ULNCY CURVES
443
asymmetrical curves include distributions of the stature of boys
and young men, in which the ranges are considerably larger
(great enough to show appreciable dilFerence between the two
projections) and in which the curve becomes symmetrical on a
log-A; scale. The weights of the adult men likewise showed
moderate asymmetry which disappeared on the log-^ scale,
again because of a greater range (from 100 to 250 pounds). In
all these cases, the class or group intervals are even and regular,
and yet from the organic nature of the phenomenon — human
growth in height or weight — ^we could suspect the desirability
of the logarithmic projection which is so successful in fact.
The true significance of the success of the log projection in
producing symmetry, when it does so, is that the proper units of
growth are magnitudes, such as those by which we measure
star-brilliancy or musical pitch, rather than increments; nature
uses geometrical, not arithmetical, units.
As has already been said and as may be deduced from the
above, it is not always necessary that both axes should be upon
the logarithmic projection. Frequently one has use for a chart,
the scale of which is arithmetically projected along one axis
and logarithmically along the other. Thus we may note at
once four possible chart-fields: the plain rectilinear co-ordinates
or A;-arithmetic y-arithmetic; the logarithmic or ^-log, y-log;
and the two semi-logarithmic, ;c;-arithmetic y-logarithmic, and
^^-logarithmic y-arithmetic. The discussion has so far turned
upon the projection of the ^JC-axis scale. With regard to the
y-axis scale, the different projections obviously do not affect
the symmetry of simple curves and we must base our selection
of the proper projection either upon the nature of the
variable to be shown and the apparent appropriateness of either
method, or upon the emphasis or detail which we wish to give
to certain parts of the curve, or upon mere convenience. The
log projection of the y-scale always gives more pointed valleys
and more rounded peaks than the arithmetical projection.
Log-logs, or the logarithms of logarithms, have been mentioned
in a previous chapter, and by their use still further changes can
be effected in the contour of the curve. In short, the student
will find ample means in these projections to study his data in
various forms in the course of his analysis.
Chapter XXXVIII
LOGARITHMIC OGIVES
It is in regard to tlie ogive that the projection of the y-scale
becomes important. It will be recalled from the chapter on
ogives, that these charts show the cumulation of frequency
series. It will also be recalled that the frequency series can be
cumulated from either end, forming either a ^^more-than’^ or a
‘^less-than’' cumulation. Hence, even on an arithmetically
projected field we can always have two ogives for the same
original uncumulated frequency series. If this series be a sym-
metrical one, the two ogives will mirror each other on both axes;
but if the original distribution is asymmetrical, they will not
mirror both ways even when arithmetically projected. But by
the use of logarithmic-scale projections, either on one or the
other or both axes, we can sometimes produce mirroring again,
a condition which often indicates that on such scales the curve
would become symmetrical. For the treatment of the ogive,
then, as for the simple curve, the proper projection of the ^-axis
is important. But in ogives we have, of course, only one
maximum and one minimum, with the entire range of the series
distributed between. Hence it is sometimes possible to secure
in ogives what can never be secured in uncumulated frequency
series — a straight line. The single exception to this is the
/-shaped frequency curve with only one tail, and this curve
will often, as has been said, be found to be an ogive in disguise,
the cumulated nature of its data being not immediately
apparent.
Now just as symmetry is more desirable in general than
asymmetry, so a straight line is in general more desirable than a
curve. For it still further simplifies the significance of the
chart, and it still further displays regularity of behavior in the
phenomena. Hence in work with ogives, which are merely
irregular curves, typically of an S-shape, the search for a
straight line is legitimate. And at this point, the value of the
444
LOGARITHMIC OGIVES
445
log-projection of the y-axis scale comes in. For it may be that
an ogive of very great curvature will straighten out into at
Yearc Total
DURATION OP STHITCBS
Number of Strikes ended in Specified Periods of Time or Lead
United States
1915-1921
( Note’- Strikes emitted ’becnuse lengths not reported;- 1916, 332; ISlT^ 616; 1918, 484j
1919, 301; 1920, 437, and 1921, 262.)
(Source - Monthly Labor Sevie'^")
e-ift 05 <0 je
StnO'<l*2>Ou> CJiSS
e-c-QooococbO OO^
»-l 0«D Ot-^tOtOr
(T> ‘a
O 0<-t csJi'OeOfC'd*-^
in to r^toa><»
CIIJRTS JND GRAPHS
446
least a close approximation of a straight line when the y-scale
is made logarithmic. This possibility applies in general to all
truly /-shaped curves. And when you have been unable to
make a curve symmetrical, you still may succeed in making one
of its ogives into a straight line, and so pull success out of
defeat, and bring order where chaos was.
A spectacular example of this close approach to a straight
line on the part of an ogive is the ^‘more-than’’ cumulative of
the distribution of personal incomes in a community. Govern-
ment reports show tabulations of the number of persons who,
according to their income tax statements, enjoyed incomes
between specified limits. By cumulating this distribution so as
to get the number of persons enjoying more than each specified
amount of income, there is obtained the data for an ogive
which will pass through such excessive ranges in both variables
as a dozen persons enjoying more than ten millions of dollars
annual income and two millions of persons enjoying more than
a thousand dollars of income. The curve of this cumulated
series will, on arithmetically-projected scales seem asymptote
to both axes, hugging them so closely throughout its length
that were the chart-field as large as the side of a house, the
curve would never leave the axes by more than a few inches.
But plot the same data on logarithmic paper and the ogive
comes so close to a straight line that one is tempted to ascribe
its variations to errors in the data. This particular example
not only illustrates the tremendous compression of large num-
bers on a log-scale, but it also nicely exhibits the analytical
power of the logarithmic method, for by its use the Italian
economist, Vilfredo Pareto, was led to formulate a ^^law’’ of
the distribution of wealth and income which was simply the
mathematical expression of the slope of the straight line. And
while Pareto’s law has not withstood the waves of debate which
it has occasioned, the straight line on which it was based
remains the best means of analysis of comparative income
statistics for different communities.
Another advantage in the logarithmic y-scale, and one
which applies to all frequency curves, as well as to ogives, is
that by its means, very dissimilar (as to amplitude or height on
the y-scale) frequency curves can be compared. Thus when
two series would on ordinary co-ordinates lie so far apart, the
one high above the other, that they could not profitably be
shown on the same chart without using different vertical scales
PERSONAL INCOMES
Huabor of Persons Reporting Incomes in Excess of Specified Aaotanta
United States
1919
(Source;- Collector of Internal Revenue)
448 * CHARTS AND GRAPHS
for them, it is often a very great saving in labor, as well as an
assistance in analysis, to use the logarithmic vertical scales.
The slopes of the various parts of the curves may be compared
upon this projection and significance attached to parallelism,
just as in historical rate of change curves. Commonly, the
method is most useful when the two curves lie upon the same
portion of the horizontal scale. Curves can also be made com-
parable by the use of percentages in the place of the numbers,
each value being turned into a percentage of the total of the
series. This requires more computing, but is for some pur-
poses superior to the use of logarithms or logarithmic y-scale
projection.
In the construction of the logarithmic frequency curves, the
principles laid down under amount-of-change frequency curves
apply as to the positioning of the field. When the ogive is used,
there should be room above the field of the chart for the original
data to enable reading from the independent variable, and it
may often be well to leave room to the right of the chart for
derived secondary data in the form of readings from the depen-
dent variable. As for other frequency curves, the selection of
the independent variable is sometimes difficult and often merely
a matter of whim, choice, or convenience. In such cases,
particularly, both the original and the secondary derived data
are useful, for you cannot be sure in advance which data you
will ultimately adopt. There is, however, in the logarithmic
frequency curve, little use for the staircase form of plotting, as
the areas between ordinates have no significance, and we can
limit the discussion of the logarithmic curve to frequency
polygons and smoothed forms of curves. As to the logarithmic
projection of scales, of course, the principles laid down in the
previous chapters apply. And other things being equal, the
logarithmic projection of the a;- and y-axes of each chart
should be upon a common scale, that is, with decks of equal
size, whatever the calibration may be.
The possibilities of application of the logarithmic frequency
curve and the logarithmic ogive are very great and the student
will soon discover that they exceed in usefulness for research
purposes the ordinary amount-of-change frequency curves as
much as the historical rate-of-change curves exceed the his-
torical amount-of-change curves.
We have mentioned population distributions as obviously
calling for logarithmic projection. This is important in sales
LOGARITHMIC OGIVES
449
analysis and merchandising research. Rent statistics, pos-
sibly an even better index of local buying power than income
statistics, have been found, where they have been compiled,
to behave as do the incomes. In building statistics, it has
been found possible to set up normals of new building for each
community on the basis of size of population, and the com-
parison of the actual building with this normal affords a useful
index of local business conditions, the entire analysis being
carried out on logarithmic paper. The possible useful applica-
tions of this form of chart are innumerable. In the engineering
world it is in common use, being much better known than the
semi-logarithmic, historical rate-of-change curves. Though
less popular than the latter and perhaps less often required, the
logarithmic frequency curve should play an important role in
the business or economic research laboratory.
PART IV. SPECIAL ANALYSES
Chapter XXXIX
THE NORMAL CURVE OF ERROR
The statistical authorities usually make a great to-do over
frequency curves, for much statistical work of a precise nature
involves the close study and measurement of frequency distri-
butions. The fundamental conception in such work is one
ideal or theoretical form of distribution which is known as the
^^normal curve.’’ The analysis is more often than not directed
at the question of whether given distributions conform to this
normal, and if not, how closely they approximate it. In
graphics the question is whether a given frequency curve can
be found to be identical or nearly identical with a corre-
sponding normal curve. For reasons which will presently ap-
pear, the discovery of this identity is always attended by re-
joicing and relief, similar to the discovery that a historical
curve conforms to the law of organic growth. For in each case
a condition of meaningless irregularity has been replaced by
the establishment of a definite and significant law.
If you select at random a hundred men, provide them with
yard-sticks, and let them measure the length of the city block
in front of your house, you will be disappointed if you expect
a unanimous report from them. Their measurements will vary
through a considerable range, and “bunch up” most thickly
about midway between the extremes. Pick your men for ex-
perience and ability and you may expect the variation to
extend through a shorter range, that is, the highest and lowest
estimates will be nearer together, but the variation will still
be present, with its bunching up at the mid-point. Provide
the men with accurate engineers’ chains instead of wooden
yard-sticks and you will still further reduce the range of varia-
tion, perhaps down to inches instead of feet or yards, but it
will still be present. The results are not changed if, instead
of a hundred men, you use but one man, letting him measure
the distance a hundred times over (provided he does not
450
THE NORMAL CURVE OF ERROR 451
voluntarily repeat his estimates). As a matter of fact, all
these measurements are approximations; the actual distance
itself has not changed or varied in the least. We are simply
confronted with the human equation and its inevitable errors.
And the interesting thing is that these mistakes or errors have
been found to group themselves in a certain characteristic for-
mation or distribution. In any sujfficiently large body of ob-
servations the scatteration or dispersion of items due to mis-
takes or errors of observation, falls ever into or approximates
the same characteristic form. Plotted upon a chart, this form
is the normal curve, and the normal curve is therefore often
called the curve of error or the curve of errors.
Under the name of ‘^the probable error,’’ artillery officers
study the scattering of gun-fire, for no two successive shots
from the best cannon in the world will hit at precisely the
same spot. If you step outdoors and pick (without choosing)
a hundred leaves from a tree you will find that while they may
be of approximately uniform size, yet there will be minute
variations of length or breadth in these leaves. Or glance at
the stock exchange quotations and arrange the first hundred
into a frequency series. In every case the form of normal
curve will again be approached by the curves of your observa-
tions. Hence the normal curve is often called the ^^probable
curve” or the curve of normal probabilities.
In algebra every school boy knows the expansion of the
binomial {a-\-h). The square of the binomial is +2ab -{-IF) .
Its cube is {a^-{-3a?b-{-'ial>^-\-h^)y its fourth power is {a^-{-
4:a^b -{-6d^F -{-^alF -{-F ) its fifth is {a^-{-Sa^b-{-10a^'^-{-lQa?b'‘^-{~
Sab^-{-b^)\ and so on. Note that these coefficients increase
always as they approach the center from either end. If we
were to plot them as a frequency curve we should again find
a suggestion of the normal curve. Carry the expansion out
into higher powers and the approximation becomes closer.
If the expansion could be made indefinitely great, the curve
of the coefficients would precisely conform to the normal
curve and for this reason the statistical study of the normal
curve is closely tied up with the binomial theorem. If in no
other way, the normal curve could be mathematically com-
puted by its means.
Enough has been said to show the importance of the normal
frequency distribution. Let us therefore attempt to describe
its appearance. This is not an easy task, in as much as any
452
CHARTS AND GRAPHS
THE NORMAL CURVE
Ordinates of the Uncumulated Series
(Formula:
(Note: Values (ordinates) are given for parts of the range (abscissae) on both sides
of the Median (origin), the units of measurement for abscissae being the Standard
Deviation, and for ordinates, the Median.)
Negative Side of Median (origin)
Positive Side of Median (origin)
Abscissa
Ordinate
Abscissa
(x)
(y)
(x)
0
1.000,0
0
-0.2
.980,2
0.2
-0.4
.923,1
0.4
-0.6
.835,3
0.6
-0.8
.726,2
0.8
-1.0
.606,5
1.0
-1.2
.486,8
1.2
-^1.4
.375,3
1.4
-1.6
.278;0
1.6
-1.8
.197,90
1.8
-2.0
.135,34
-2.2
.088,92
2.2
-2.4
.056,14
2.4
-2.6
.034,05
2.6
-2.8
.019,84
2.8
-3.0
.011,109
3.0
-3.2
.005,976
3.2
-3.4
.003,089
3,4
-3.6
.001,533,8
3.6
-3.8
.000,731,8
3.8
-4.0
.000,335,5
-4.2
.000,147,75
4.2
-4.4
.000,062,52
4.4
-4.6
.000,025,42
4.6
-48
.000,009,930
4.8
-5.0
.000,003,727
5.0
Fig- 369.
change of scales upon the chart, along either axis, results in a
change of its shape. Enlarge the horizontal scale and the
figure of the curve is spread out wider, reduce that scale and
it becomes narrower. Increase the vertical scale and every
part of the curve is raised, making its peak much taller; dimin-
ish the scale and it is flattened out. Hence the normal curve
may have an infinite number of conformations. Always, how-
ever, it has a hump at its horizontal mid-point where the fre-
THE NORMAL CURVE OF ERROR
453
quencies are thickest. Hence it may be described as bell-
shaped, showing a peak at the mid-points and a die-away curve
or tail at each side of the peak, one at least of the tails being
asymptote to (i.e. approaching but never reaching) the hori-
zontal axis.
Now there are any number of possible curves which fit the
above description but are not normal curves, hence the de-
scription is not a definition. Curves may be rounder or flatter
or more pointed at the top or may slope away at different
angles from the corresponding normal curve or any similar
normal curves. Hence it is important to be able to distinguish
between normal curves and curves which are not normal. It
is not enough to find in analyzing a given distribution, that its
curve has a central peak and is symmetrical. The student
should be familiar with the various forms of the normal curve,
but it is not always possible, even for an expert, to be quite
sure whether a given curve is normal or not, if he can rely only
upon inspection. We need to compare the given curve with
a precise drawing of the corresponding normal curve. And re-
membering the infinite number of normal curves, it becomes
difficult to select the proper one for the given distribution.
Obviously the compared normal curve, that is the ideal dis-
tribution, for a given series, is one which will have the same
total area under the curve, that is, the same number of obser-
vations or items in the series. But there is an infinite number
of normal curves with the same inscribed area, differing from
each other according as their peaks are tall and narrow or
short and broad. So the proper normal for any series must
be one which will intersect the given curve at certain points.
Usually the standard deviation is used as the abscissae of
these intersecting points. So we are led into a tedious math-
ematical process, only a part of which is the computing of
what statisticians call the standard deviation and all of which
is more statistical than graphic. The comparison of a given
curve and its normal is not easy by this method.^
In the next chapter will be described an easy trick by
which you can make such a comparison graphically, and learn
whether the given distribution or series is normal or not, and
if not, how closely it approximates the normal distribution.
^ The two ways of fitting the proper normal to a given curve, other than the
graphic one in the following chapter, can be found in Yule, Theory of Statistics, pp.
307 - 9 .
Chapter XL
PROBABILITY CURVES
Those readers who have understood the form of graphic
legerdemain by which the adherence of a historical series to
the law of organic growth is flashed upon the rate-of-change
chart paper, will be prepared for a similar trick by which the
adherence of a frequency series to the normal distribution is
graphically shown. They will anticipate that the graphic
form eliminates practically all computing and calculating
from the treatment of the data, this computing having been
absorbed once for all into the projection of the scale of the
chart. The trick is very simple. It consists of plotting the
ogive of the distribution upon paper which has been specially
ruled off in such a way th at the ogive of any normal curve will
become a straight line upon it.
The normal ogive, as you know, is S-shaped. If you
know the ordinates of the normal ogive you know precisely
how much to distort the scale for the dependent variable so
as to produce a straight-line projection of the normal ogive.
You need only select points equidistant along the vertical
scale, read the abscissae of corresponding points on the ogive,
lay off these horizontal values vertically (on the vertical
scale), and shift the scale figures from the old to the new ver-
tical scale. By doing this you have made the ordinates and
the abscissae of each point on the normal curve alike and so
of course the normal curve becomes a straight line. But any
other S-shaped curve, any other ogive, or any curve at all,
which is not a normal one, will fail to straighten out perfectly.
And so at a glance you can see, by this chart, not only whether
a given distribution is normal, but if it is not normal, how
closely it approximates or deviates from a normal distribution.
To plot a given ogive upon the probabilities projection of
the dependent-variable scale just described, it is best to turn
all frequencies, i.e. dependent variables in the data, into per-
454
THE NORMAL OGIVE 455
Ordinates of the Cumulated Series
(Note: Values (ordinates) are given for parts of the range (abscissae) measured in
both directions from the Median as origin, the units of measurement for abscissae
being the Standard Deviation, and for ordinates, the total of the series. The table
can also be used to show fractions of the area under the normal curve (uncumulated)
lying on each side of verticals (ordinates) from specified abscissae.)
456
CHARTS AND GRAPHS
centage figures. For the peculiar spacings of the probabilities
projection are arbitrary and cannot be freely changed. The
total of a series is the limit of its cumulation, and the cumu-
THB PROBASILITISS FfiOJECTICN
fhowlns th* nsthod of censtructlng «
probabilitio# proJocUen, along tho Y-sJtls.
Cl.«»t-than
CunulatlTa)
Percantaga
of
Fraquoneiaa
Fig. 371.
Showing how the dependent y-scale of percentage-frequencies is readjusted from
the arithmetical projection which gives the curved ogive to the probabilities
projection which gives the straight-line ogive. The frequencies must invariably
be converted into percentages for this chart. ^
lated series has definite limits (for its dependent variable),
these limits being “no items” and “all items,” that is, 0 and
100% of the total of the series. Hence the only common
measure for all frequency series is a percentage one and the
normal curve and ogive are both given in all tables in per-
PROBJBILITY CURJ'ES
457
centages. The distortion of the dependent variable scale is
made for values of these percentages, and you must not seek to
shift the scale figures of the probabilities projection scale as
you could an arithmetical or logarithmical projection scale.
Your only way to alter the probabilities scale is to turn the
percentages into values of your particular series, in which case
the scale is restricted to this series, a fruitless if not danger-
ous step. It is sufficient to turn the cumulated series into
percentages before plotting upon the probabilities projection.
.11) O© « H O O<De*-OU>^,«0 « H O rj N «0 .tirtiOC;®® O H Cl W^lOO COOinft
(m Wr-f f-iHH*** • • t * » r-t H H HNOlw
I It I I » I • 4 1 I Kang®
Figr. 372 , A Less Useful Form in Which the Independent or x-Scale
of the Range is Readjusted to Straighten Out the Ogive.
It is less useful because the range-intervals must be turned into units of the
standard deviation before plotting. Its sole advantage is that the frequencies
need not be turned into percentages.
The scale for the independent variable, that is, ^-axis
scale, may be projected either arithmetically or logarithmically.
If a given series will straighten out upon the former, it indi-
cates that its curve would be symmetrical upon an arithmetic
projection; if the ogive straightens out upon the logarithmic
paper the distribution is one of the asymmetrical types which
are made symmetrical by a logarithmic projection. Hence the
probabilities paper not only gives the approximation to the
normal, but it also determines whether that normal be sym-
metrical upon a geometrical or arithmetical basis. Chart
458
CHARTS AND GRAPHS
Fig. 373. Commercial Probability Forms.
These twQ sheets, an arithmetical-probability form and a logaiithmic-probability
form, are marketed under these names b^ the Codex Book Company. — Per-
mission of the Codex Book Co.
forms with the probabilities projection of the dependent vari-
able scale, are published with both projections of the inde-
pendent variable scale.i
^ The true nature of the “dependent” and “independent” variables in ogives has
already been discussed. (See Chapter XXIX.) The same considerations hold of
the ogive on probabilities paper, but by accident some of the publishers of probabil-
ities paper have reversed the arrangement in their printed scales, thus inadvertently
hastening the time when, in the author’s opinion, statistical practice will correct
itself and plot :v-frequencies, y-range.
Farm Acreaga
PROBABILITY CURVES
459
The probabilities scale, calibrated in percentages, will never
reach either zero or one hundred per cent in either direction.
SIZE OF FAHUS
P®rc»nt Diatrlbution of Pams of Specified Number of Acres or Greater
United States
1890-1920
(Source:- United States Census)
Fig. 374. The Logarithmic Probabilities Projection in Use.
Note the close approach to straight lines. Also the ease with which, median,
quartiles, etc., and interquartile range are found.
for the true normal curve is asymptote to the x-d.xis and its
100% parallel. We can therefore never plot the two limits,
460
CHARTS AND GRAPHS
0 and 100%, on this paper. We can only come as close as we
wish toward these limits by extending the scale on into the
coLLne sAunins
, Sftl&rlas In toorlMn Colle'se UtilTrarsleiaa
IneXudlng public and Private Institution*
nnltai Stitss
1920
{Source - United Stele* Bureau of EdueaHon)
Fig. 375. An Arithmetical Projection of the Dependent
^ (or Frequency) Scale.
small fractions near each limit. After we pass the points of
.01% and 99.99% the scale resembles a logarithmic projection
so closely that we can add to it by logarithmic scales. But
the tails of a curve are least significant, and it is rarely worth
while to make this extension. Unless we deal with large series
containing over ten thousand items and so grouped that the
terminal groups have only one item each, we shall not need to
plot points less than .01% or greater than 99.99%, Usually
PROBABILITY CURVES
461
it is safe to chop off all of the scale beyond 1% and 99%, thus
reducing the chart, either because of the absence of data out-
COUESE SAUfilES
In Aaarlcan Collnfea aid Onlvaraitiaa
Including public and PriTBta Inatlluticna
United Stataa
1920
(Source • United States Bureau of Education)
64 ITS 628 lOlO 840 TT5 1040
ISO 780 IISO ITSO 1250 14S0 2200
250 1050 18D0 2000 1800 1750 2600
300 1120 1870 2100 1720 1900 2800
570 1200 1640 2200 1880 2100 3000
800 1340 1750 2380 2100 2380 S4S0
«30 1800 1900 2500 2400 2700 3900
740 1570 2040 2620 2670 5000 4600
800 1700 2200 2760 5080 3400 8200
1000 1760 2300 2860 3300 3660 6600
1080 1600 2400 2950 3850 4000 8000
1380 2000 2650 3200 4280 4800 8600
2000 2800 3400 4000 6000 7750 9200
Fig. 376. A Probabilities Projection of the Previous Chart.
Note the secondary data and bar-chart obtained from interpolation of the curve
at the various decils, showing the median and other salaries for each group of
educators. The swing of the upper tails of the ogives away from the straight line
of the normal distribution might be interpreted as due to a few institutions
which give titles out of proportion to the salaries attached thereto,*
462
CHARTS AND GRAPHS
side this range (other than the limits) or because of the lack
of their significance.
The significance of the ogive is always elusive to the layman
and the probabilities projection of the ogive, while it clears
up, for the technician, the question of normality, is even more
baffling in other respects. A little study will show us, how-
ever, that both the direction of the curve and its position are
significant. When comparing two ogives, if we find the ogive
of one series further out along the horizontal scale than the
OOTHTT OF FACTORIES
Th« Value of Products of Manufacturing Establishment a
.having less than specified value of products
per establishment
United States
1904-1914
In percentages of the total
Source;— U S Census
Fig. 377# Symmetrical so Far as Data Obtains.
t
other, it means that the items in that series are greater than
corresponding items (in similar parts of the distribution) in
the other series. Thus an ogive for the heights of children
PROBABILITY CURVES
463
will be to the left of an ogive for the heights of adults. If the
two simple frequency curves had been plotted they might
overlap but the main bodies of each would be at different
positions along the scale. The significance of the positions of
ogives will be clear if the corresponding simple curves be
miOLESALB PRICE CHANOSS
Qlatrlbutlon (by aagnituda) of "Chain Relatives" of Rholasala Prioae of 23C Coamoditla*
Onited States
1891-1913
(Total nunbar of chain ralativea showing change of price, 4,881, 697 cases of no change being omitted)
(Arranged from Ultchell "Index Humbers of Wholesale Prieea")
(ifean * 1,51 Ircre».se, Sta-^dard De/let. cr ■ 14.44^1. dotted line shows fitted normel)
imagined underneath them or if horizonal bars be imagined
as lying between each ogive and the y-axis.
As in other ogive-charts, secondary data may be derived
from the curves, being the reading upon the ^c-axis scale of
intersepts of the curve and the horizontal rulings. In other
words,' while the curve is drawn by plotting the ordinates at
certain abscissae we may interpolate from it the abscissae of
certain ordinates. The readings are usually taken at the
464
CHARTS AND GRAPHS
median and quartlles, and decils, and occasionally at the ex-
treme percentiles 2 In this derived or secondary data we see a
reversal of the dependence of the variables, the dependent one
nOBATIOJI OF STRIKES
of Strllcx andod in Spaolfiad Parlods tiaa or tool
llflltod Statoa
1918«19jSl
(Sourea - Kanthly Labor Rarlav)
Fig. 379a The Comparison of Ogives for Different Dates.
This gives a historical curve instead of a bar-chart for the secondary derived
data. The ogives are useful for interpolation but the derived curves afford the
best view of changes.
2 That the interpolation for median, quartiles, etc., does not give cases, but values,
has already been pointed out. (See Chapter XXIX.)
PROBABILITY CURVES 465
becoming independent. This secondary data affords most of
the statistical measures of both dispersion and skew.
The significance of the direction of the ogive upon the
probabilities paper (when it approximates a straight line) is
more subtle, but far-reaching. For as we have seen, the range
of observations may vary in different series. In the example
already given of measurements of a given distance, we saw
that as precision of measurement is increased, the distribution
becomes more concentrated, the scatteration less, and the peak
of the simple curve taller. It is precisely this condition which
will make the ogive curve more nearly perpendicular to the
x-axis. As the dispersion increases and the precision dimin-
ishes, the ogive curve swings about toward a horizontal direc-
tion. These considerations hold as well for the ordinary ogive,
but are more clearly seen and measurable in the ogive pro-
jected upon probabilities paper.^
In the analysis of frequency data by means of the ogive
curve upon the probabilities projection, a difficulty is often
met in that the data is incomplete, no figures being obtainable
for a portion of the range. Thus the statistics of income do
not include the personal incomes below two thousand dollars
for heads of families and one thousand dollars for individuals,
and for this reason omit perhaps ninety per cent of the popu-
lation. Astronomers have estimated the number of stars of
each magnitude down to the twentieth, but do not carry their
estimates much further. In such cases as these we do not
know the entire ‘^population,^^ ^^universe,” or total body for
which the distribution applies, and hence cannot turn fre-
quencies into percentages.
The problem is largely statistical but affects the subject
of charts in that an ogive upon probabilities paper, of these
incomplete distributions, obtained by turning the frequencies
into percentages of the known sub-total, will almost certainly
fail to form a straight line although the entire distribution
may be perfectly normal. It is not legitimate to plot such
parts of distributions upon the probabilities paper unless we
can turn the frequencies into percentages of the true total. ’
At this point, therefore, we will mention a statistical trick by
which you can sometimes dodge the difficulty. For in all fre-
quency data there are two possible series, both covering the
3 On the probabilities projection, the mode can not be graphically determined by
'the slope of the curve, as it could on arithmetical projections of the ogive.
Relative Number ct Stars of Various Ka^nitudes and Quantity of Licnt Thrrofrom
(Wore-than cumulativea only. JPull llnpa for plots as ppr scales shown, dotted
lines for arithmetical projection of scale for relative brilllat»cy. }
Figr* 380. The Ogive of the Units of Measurement of the Items (Star
Brilliancy) in an Incomplete Series is Straighter and more Reli-
able than the Ogive of the Items (Number of Stars).
See footnote on opposite page.
PROBABILITY CURVES
467
LABoa rtni
Separated Eisployees and Equivalent wuaber of Full-yoaf Jobs, Subject to Instability*
clasaafiad as to length of service.
("Enployees” • percent distribution of 2,581 separated enpleyoes)
("Jobs’* • percent dlstrib-jtion of aggregate length of tlta ser/sd bj 2,563 separated enployooa
who eerved less tnan five years)
Sugar Refinery
California
Year Ending Hay 31, 1918
(Scu’^ro.. Paul ?. Snesendoa'
Jobs 9J.0 97.3 92.6 75.7 61.4 43*0 25.7 19.3 0.(
Esployoes 78. 65. 46. 21. 12. 6. 3. 2. 1,
Fig. 381. Another Example of the Two Interconvertible Frequencies
for the Same Data.
same observations. These two series have been described in
an earlier chapter. They are, briefly, the count of items, and
the count of units of measurement of the items, group by
group, through the distribution. The point is that these two
series for the same phenomenon are oiFten available and can
usually be estimated if not available. And if a considerable
portion of the data of one series be missing, it is generally
true that the missing portion covered items of small or negli-
gible numbers of units individually. If then we convert the
Note to Fig. 380
The dotted lines are plotted upon an arithmetical scale (not shown on the chart)
of relative brilliancy. The full lines are plotted by the two horizontal scales
(appearing on the chart), namely a logarithmic projection of relative brilliancy
and an equivalent yithmetical projection of magnitudes. It would also have
been possible to project the magnitudes logarithmically (thus obtaining a log-log
projection of brilliancies). The interesting point is that star magnitudes though
projected arithmetically are in themselves geometric units and form a logarithmic
projection of brilliancies.
468
CHARTS AND GRAPHS
item data into unit data, we generally find that the importance
of the missing portion of the data has greatly diminished. Thus
while income statistics omit perhaps ninety per cent of the
families and individuals in the country, they omit only about
ten per cent of the total income. In the case of stars, the
astronomers have computed the light of all stars, so that the
unit data is complete, while the item-data is incomplete.
When the incomplete data can be reduced to a relatively small
amount in unit-data, it may more safely be estimated, and
so completed. Thus by converting the data into another
form, we may find it possible to project the ogive curve upon
probabilities paper with satisfactory results.^
OUTPUT OF FACTORIES
The Value of Products of groups of Employees
eraproyed in Manufacturing Estahlishments
(all groups composed of employees in establishments
having lonrost value of products)
United States
1904 - 1914
(in percentages of total)
Source: U S Census
Class of
Establishment
by Value
of Products
1904
1909
1914
Employees
Huniber
Product
Value
Employees
Number
Product
Value
Employees
Humber
Product
Value
Lese-than |5,CX)0
1*9
1*2
2*2
1*1
1.8
I.O
Less then 120^000
9*6
6*3
9.3
5*5
7.9
4.V
Leas than llOO^OOO
28*4
20.7
25.8
17.8
22.1
16*2
Less than ♦1,000,000
74.4
62*0
69*6
56.2
64.6
61.3
Any value whatever
100.0
100.0
100,0
100.0
100.0
100.0
Fig. 382. Alternative Data Yielding the Lorenz Curves.
^ “If the observations are not complete (i.e. cover only a small part of the unknown
total range of the variable, though it is highly desirable that what observations we
have do not constitute a mere extreme tail), it is possible to fit a normal curve to
the data by means of fitting a second degree parabola, by the method of least squares
{y = to the logarithms of the number pf observations in each interval.
The theory is based upon the equation of the normal curve,
Taking logs of both sides, y—Kie
log^- y^logfiTi— -i
K.2
we get a second-degree parabola.” — Dr. Frederick R. Macauly.
PROBABILITY CURVES
469
We have spoken of the alternative series into which an]!-
distribution may be converted. The combination of these two
yields, upon arithmetical paper, the Lorenz curve. And it is
for the Lorenz curve, of all types, that we can profitably use
double-probabilities paper, that is, paper projected upon the
probabilities scale along both axes. For, as will be remem-
bered, the tails of a Lorenz curve are nearly asymptote to the
axes when the dispersion is great, and the values near either
extreme become difficult to interpolate or read from the chart.
Moreover, when the groups or classes, into which the distri-
bution has been arranged, are few, the curvature of the curve
becomes angular and interpolation is unreliable throughout
OUTPUT OF FACTORIES
The Value of Products of groups of Enplcyoes
eiiiployed in KlanuTacturing Estubli shnieiits
(al3 groups composed of emplcyoes in estahlisliments
having lowest value of products)
United States
-1904
1909
1914
Source; U S Census
(In percentages of the total)
percentage of aggregate Numher of Employees
Fig. 383. The Double-Probabilities Projection Straightens Out the Lor-
enz Curves When of Normal Distributions.
the length of the curve. But when both axes are ruled on the
probabilities scale, the tails become indefinitely long, the zero
PERSONiiL INCOMES AND TAXES
Distribution of Incorae and tax among tax -payers
United States.
1919
(Source:- Collector of Internal Revenue)
Percentage
CD ID CO o>
o> o> Cf> o o>
• CJ> 0> CJ> CJ>
Fig. 384, Double-Probabilities Projection for Several Lorenz Curves.
The chart shows that, for example, one-half of the tax-payers paid only 3% of the
taxes, having only 22% of the income, so that one-half of the income of tax-
payers yielded only 8% of the taxes. The curves are not straight because the
tax-payers do not constitute the entire income-receiving population and form
therefore an incomplete or truncated part of what is probably a normal dis-
tribution*
PROBABILITY CURVES
471
and hundred per cent points disappear, receding to infinite
distances, and the curve, throughout its length, becomes very
close to a straight line. When the distribution is normal, ob-
viously the curve becomes a straight line, and hence the
straight-line Lorenz curve on double-probabilities paper is a
quick and useful indication that the distribution is normal.
Obviously the accuracy of interpolation is improved by this
straightening out of the Lorenz curve, as is also the facility
for detailed comparison, such as through light-analysis, of
several distributions so plotted.
The utility of the probabilities projection must be ap-
parent to those who deal with frequency data. For analytical
purposes, as a labor-saving and illuminating chart, it takes its
place beside rate-of-change paper for historical data.
Chapter XLI
SHIFTED ZERO-POINTS
We have seen, thus far, three great types or kinds of scale
projections, arithmetical or uniform, logarithmic or geome-
trical, and normal or probabilities. We have seen these com-
bined in every way upon the two axes of the chart. All
this has been done in the search for simplicity or regularity of
behavior, and convenience or ease in interpolation. There
remain still other projections of the chart-scales, which serve
the same purposes and will be discussed in later chapters.
And there are a few minor variations of the logarithmic pro-
jection which can well be discussed here.
We have seen that historical data can be plotted with either
logarithmic or arithmetical vertical scales, but that it is not
correct to use anything except 'an arithmetical scale on the
horizontal axis. To this rule we may now note tw^o exceptions.
The first arises in the case of data with a definite origin point.
Thus the pseudo-historical frequency series which involve time
have already been put upon logarithmic A:-axis scales together
with other frequency series. From these frequency series in
which time is the independent Variable, it is but a short step to
strictly historical series, in fact the distinction disappears here,
the same series being called equally well a frequency or a
historical one. But there is also a class of purely historical
data, involving specific points of time, which can be placed
upon a logarithmic A;-axis. This is data, generally of a geo-
logical, or other scientific nature, covering very large periods of
time, such as the age of the earth, and its important geological
eras.
The second variation of historical series is extremely
interesting, though of very limited application. It may be
called the retrospective projection. If from any point of time
we look backward over the years, we may notice that the more
recent events stand out more clearly, and in more detail, while
472
SHIFTED ZERO^POINTS
473
the events of early years become more vague and their details
lose importance. In business, this importance of the last years
is recognized and business statistics therefore often contain full
detail for the most recent period and only brief summaries of
previous periods. In histories, the space devoted to ancient,
medieval, and modern times, usually shows a similar com-
pression of earlier times. If these witnesses are of any value,
they testify that the importance of detailed data diminishes as
its remoteness in point of time increases. And the chart-maker
has therefore a legitimate object in devising a chart method to
display the data in its proper detail or lack of detail. .
Several methods have been tried to meet this charting need.
By the silhouette bars a few facts of the past history are given
in addition to the very latest figure. By the juxtaposition of
two curves with a single y-axis scale but different ;v-axis scales,
one, let us say, for years, the other for months, data for a recent
period can be given in full detail, and that of a previous or the
entire period in summarized form. But the inventive mind
will seek still a better method, which will not have the rigidity
of the last and in which the disappearance of detail will be
gradual and so we arrive at the use of a logarithmic A:-axis scale
projection, reversed in its direction so as to compress the earlier
periods of time upon the chart.
This retrospective logarithmic projection of the time scale
has both intriguing advantages and baffling disadvantages.
Upon its credit side we may observe that it presents precisely
the degree of importance to data at various points along the
line that we desired and has unlimited possibilities of extension
backwards into remote antiquity without consuming space
wastefully. If we are, in the year 1900, let us say, to look back
over the centuries, we shall doubtless attach the same relative
importance to the entire nineteenth century as we do to the
seventeenth and eighteenth combined, the same relative impor-
tance to the last thousand years as to the two previous thousand
years. Important exceptions occur, of course, but in the main,
this proportion of weight of importance holds, else historians
would not be justified in devoting their space to the different
periods in these ratios. . The real nature of that phenomenon
of change which we call the passage of time is still a profound
mystery and it is an interesting speculation that it may in some
occult way combine elements of progressive and regressive
organic growth. But idle as this thought may be, the chart of
474
CHARTS AND GRAPHS
a m
MNABtocM osKise tra-dc-Qiiicm BMiai>«r« la tlw oat«r amtrfitai
igi3-X92X
(SoorM:* ltot(th]jr Uibor Srrlttj
Fig. 385. A Historical Retrospect with Reversed Log Plotting for the
Horizontal or Time Axis.
The purpose is to present recent developments in greater detail.
the geometrically retrospective historical curve has a certain
value in presenting graphically a survey of the past in proper
emphasis and detail, and alFording a comprehensive picture not
otherwise equalled. Owing to the fact that the significance of
the slopes of the curve disappears in this chart, its usefulness in
mathematical curve-analysis will always be limited, if not
doubtful. But when the gun-shot method of plotting be
employed (that is, isolated points be plotted) its success is
marked. It is not improbable that in time all school histories
will be illustrated with diagrams in which events will be entered
at their proper positions upon such a scale.
The chief disadvantage of the logarithmic retrospect chart
lies in the fact that before entering the time-figures upon the
scale, these figures must be computed back from some origin
point, either in the present or in the future. The choice of
origin-point for our backward count of the months or years
directly effects the degree of expansion which the most recent
periods of time will undergo on the chart-scale. If we take the
SHIFTED ZERO-POINTS
475
HEOLBSiLB' mticss 01 tE> xms>
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Fig, 386. Another Example of the Same.
immediate present, then last month's figures will be plotted at
unity, as being one month back, the previous month's data will
be entered at the logarithm of two, on the left side of unity, and
so on backward along the axis of the chart. But where will we
plot next month's figures when they appear.? We cannot plot
them at the logarithm of zero. In other words such a retro-
spective chart would have to be redrawn each month. The best
way to avoid this is to assume an origin point of time some
distance in the future. Although this destroys the significant
relation of the more recent events, so far as that relation was
desirable, yet it enables us to bring the chart up to date for
sorne time to come. But the student will now see that com-
parisons cannot be made from one such chart to another unless
common ^View-points" have been used in both. The dis-
advantage is so serious as to make the chart useless for research
purposes, but it is still worthy of notice for general records and
popular presentation.!
^ periods of time, a ‘‘squares” or other powers projection will serve
equally well and can be continually added to.
476
CHARTS ANT) GRAPHS
The use of altered logarithmic scales for the frequency series
is a step which requires much more technical mathematical jus-
tification. We seek here to bring not detail nor convenience,
but symmetry and regularity, to the curve. Experiment will
show that a great many asymmetrical distributions can be
made to approach symmetry by assuming false origins and
correspondingly altering the logarithmic scale projection. That
this is necessary for data in which the original zero is a false or
arbitrary one, having a real positive value, has already been
pointed out. But when cause for shifting the zero does not
clearly exist in the very nature of the data, the student should
be slow to alter it, for even the most excellent symmetry which
may be induced thereby may be utterly lacking in significance.
This applies not only to shifting of zero points, but also, of
course, to reversing of directions from the zero, a step which is
closely allied in that it amounts to giving to some value above
the maximum of the range an assumed value of zero and treat-
ing the resulting negative values in the range as if they were
positive.
There is open to us another alteration of the x-zxis scale
projection which is not of any value for historical curves,
namely, the anti-logarithmic projection. For just as we have
been able to plot the logarithms of our scale-figures, so, too, we
can plot their anti-logarithms. The device has results similar
to the retrospective projection, in that it expands the larger
numbers and compresses the smaller ones. Here, too, caution
must be used in attaching significance to the results. But the
mathematical interpretation of this projection, though still
technical, is much simpler. There are in fact certain classes of
measuring units which are definitely of a geometric nature and
not arithmetical: It would be useless to plot these upon
logarithmically projected scales, for the numbers themselves are
already logarithms, and the log scale would really give a log-
log projection. Thus stars are classified as of various mag-
nitudes, each magnitude being two and a half times as bright
as the next. Musical pitch is measured in tones and octaves,
each octave having twice the wave-frequency of the preceding
octave. In these cases if we seek an alternative projection we
must obviously use the anti-logarithms of the magnitudes and
tones. Both the anti-log and the log-log projections may
occasionally be useful upon either axis of the chart,
Chapter XLII
CURVE-FITTING
The reader has now seen a variety of ways by which sym-
metry or regularity can be brought to the curves of historical
and frequency data. The reason, as he has seen, behind this
quest for simplicity, for straight lines, for parallel or mirroring
curves, and the like, lies in the ease with which generalization
can proceed from such forms, and the degree of confidence with
which we may accept the data as reliable samplings and their
curves as significant pictures. When we see the curve of the
country’s population mounting higher so steadily that it forms
one single straight line, we know, without performing any
mathematical exercises at all, that the population grows at a
constant rate, and we are thankful to the logarithmic projection
of the chart which has yielded this simplicity. When we see
the curve of our advertising appropriation paralleling, on rate-
of-change paper, the curve of our gross sales, we know without
any computing that the same fraction of the dollar has gone
into advertising every year. When we see the cumulated curve
of the nation’s population as divided into cities of various sizes,
forming a straight line upon the probabilities paper, we know
that without appreciable error or efi^ort w'e can by interpolation
find the number of persons inhabiting communities of any
particular size, whether or not the Census has mentioned com-
munities of such size. And in every case, the same causes
assure us some degree of confidence in the reliability of our
observations as fair samplings, when we have put partial data.
The question of the reliability of data is properly a statis-
tical one, for which the student should consult the statistical
authorities.! It arises in the collection of data, before charting
has begun, and only recurs again when the near approach of a
charted curve toward regularity and simplicity raises the sug-
^ See particularly Bowley, p. 178,
477
478
CHARTS AND GRAPHS
gestion that the deviations of the actual curve-line from the
desired simplicity of form are due to errors of data. Thus if
the ogive of the distribution of incomes is very near to a straight
line on logarithmic paper, it is but natural that the theory
should arise, as at least a tentative explanation, that the
deviations are due to omissions in tax collection, evasion in
income reporting, or in some cases chance variations due to few
observations.
And it often happens that in the desire to justify a theo-
retical simplicity, we are too ready to excuse deviations from
it as errors in the data or chance variations. A straight line
may fit so closely to the data that we feel sure that it, instead
of the observed curve, represents the truth. Thus the entirety
of Pareto’s law of incomes is a result of adopting the fitted
straight-line rather than the actual income curve (ogive).
Since the income curve is truncated at its lower end owing to
lack of information on incomes below the tax limits, that law is
based upon insufficient data. And recent Investigations lead
to the belief that the true ogive of incomes is not a straight line
upon logarithmic paper, but upon logarithmic probabilities
paper .2' Slight deviations from a straight line are not con-
clusive evidence of errors in the data and the student should be
extremely careful in drawing hasty conclusions from a close
approximation to a straight line upon the special projections
which have been described.
There is, therefore, always a danger that the close approxi-
mation of a given curve to a straight-line, or other simple
theoretical curve, is fallacious and deceptive. This caution
cannot be too strongly emphasized. It attaches prima facie
to all attempts to fit theoretical curves to actual ones and
casts upon him who would fit such curves the burden of proof.
The presumption is that the deviations of the given curve,
from the fitted one, are significant. The removal of this
presumption may call for all the analytical powers of the
statistician, but we should always start with the presumption,
and never lightly abandon it.
There may be many reasons why the deviations are insig-
nificant. Some of these may be found in the particular cir-
cumstances surrounding the collection of the data, such as
bias on the part of the investigators, or difficulties of observa-
2 See National Bureau of Economic Research, Income in th United State.^
CURJ^E-FITTING
479
tion. But, whatever else may be found, there is likely always
to be one cause for the lack of a perfect fit, in what are com-
monly called ^^chance variations/^ These are more marked
in small samplings than in large ones. A definite mathema-
tical law for the probability of this occurrence can be found in
books on the subject. The reader who recalls the normal
curve of error will understand the inevitability of such chance
variations. And, needless to say, when we can safely consider
the deviations of a given curve from a fitted theoretical one to
be due to chance variations, these deviations lose all signi-
ficance and we may safely proceed with the fitting.
The theoretical curve which we propose to fit to a given
curve may have any shape. The simple linear curve or
straight-line upon plain paper with arithmetically-projected
scales, is merely the simplest of these. And the reader will
remember that, in the discussion of cycles in historical data, the
fitted straight-line was called the '^secular trend.'' It is a very
crude secular trend, convenient, but in most cases not pre-
cise. The reader has since seen that, for most economic
data, a straight-line upon the rate-of-change (or semi-log)
paper would be more accurate. Still other ‘^trends" and
straight-lines will be described in later chapters. For frequency
data, the fitted curve is generally the normal curve, or its
equivalent straight-line upon probabilities projections. Of
the significance and appropriateness of these straight-lines,
in each case, the reader has already a general understanding,
and the mathematics of these and other straight-lines will be
discussed later.
While so much attention is being given to the various
charting methods by which theoretical curves are reduced to
straight lines and by v/hich actual curves are more easily com-
pared with theoretical ones, it would seem well to mention
briefly the mechanics of fitting. This problem arises after
the particular theoretical curve to be fitted has been chosen.
Let us assume that it is a straight line upon one of the chart-
forms already described, such as the semi-logarithmic or rate-
of-change paper, or the probabilities paper, or even the plain
uniform paper with arithmetically projected scales. The
problem of fitting is virtually the same in all cases.
Whenever a straight-line (in the example we have taken)
is to be fitted to a given curve, and that curve does not form
in itself a perfectly straight line, it is obvious that the straight
480
CHARTS AND GRAPHS
line may lie in an infinite number of slightly different positions
and still fit very closely to the given line. The problem is,
therefore, to find the particular straight line (or other theo-
retical curve of the selected type) which gives the best of all the
possible fits. If we call the deviations of the given curve from
the fitted one, its '^residuals,’" the problem is, broadly speaking,
to find the fitted curve which makes the total of these residuals
a minimum (that is, the least possible sum for the given curve)
There are three outstanding methods which have been
developed for determining the best fitted straight-line. These
may be called the graphical method of selected points, the
method of averages, and the method of least squares.^ Of
these, the first is the simplest; the last, the most accurate;
and the second, the most satisfactory because both fairly simple
and fairly accurate. The graphic method of selected points is
nothing more than laying a transparent straight edge or tightly
drawn piece of thread over the curve and adjusting its position
until an equal number of points appear on both sides of the
straight line and the fit appears optically most satisfactory.
The other two methods are mathematical processes, for which
the reader will have to consult the proper statistical authorities.
But the first of these mathematical processes for deter-
mining the position of the fitted straight line, namely the
method of averages, is also capable of a graphic solution. If
you join the first and second plotted points and plot a new
point midway on their joining line, this new point will repre-
sent their average. If you repeat the process with the third
and fourth, the fifth and sixth, and so on with each successive
pair, you can reduce the whole curve to a slightly shorter curve
with only half the number of plotted points all of which are
averages. On the new curve fresh averages can be plotted,
this time representing averages of averages, or averages for
four points on the original curve. After repeating this opera-
tion a sufficient number of times, you can reduce the longest
curve to a series of two average points, through which a fitted
straight line can be projected.
3 Because the algebraic sum of the differences from the mean is always zero,
statisticians often use the squares of these deviations. Strictly speaking, therefore,
the problem is to find the fitted curve which makes the total of the squares of the
residuals a minimum. If the residuals alone, instead of their squares, be used, we
must seek to make the arithmetical sum (that is, the sum of the residuals, disregarding
their signs) a minimum.
^ Cf. Merriman’s Method of Least Squares ov Bartlett’s Method of Least Squares.
CURVE-FITTING
481
For many purposes it is sufficient to fit curves by inspec-
tion, just as it is sufficient to correlate them in this way. The
graphic analysis, made more precise by "^light analysis,’’ is a
tremendous labor-saver, and may serve at least in the pre-
liminary stage of the study, at least.
In fact, curve-fitting is but a variation of correlation,
being merely the determining of the theoretical curve which
best fits or correlates with the given curve. And the con-
siderations affecting correlation likewise govern curve-fitting.
For precise purposes, the mathematical method of least squares
should be employed and a mathematical coefficient of correla-
tion and probable error be computed to measure the success
of the fit.
Chapter XLIII
SPECIALLY PROJECTED SCALES
We are about to embark upon an orgy of distortions,
modifications, and special projections of the scales for curve-
charts, all of them being designed graphically to facilitate the
study of particular data. The general principles and the
more useful forms for the non-mathematical reader will be set
forth in the present chapter. In the succeeding chapter, the
mathematics of all special projections will be discussed, from
which any particular projection can be designed. The present
chapter will suffice for most.
It will by this time have occurred to the reader that the
scale figures or calibrations may be plotted or graduated at
any points along the axis of the charts which we desire, and
can therefore be made to express any function of the variable
plotted thereon. Thus the logarithmic projection is merely
one in which, if we can designate by X the scale-figures or
calibrations and by x the actual distances at which these are
placed along the axis, then in the logarithmic projection x —
log X (and X = anti-log ;c: and 10 From this it is no
difficulty to proceed to the scale projection of other functions
of the variables. A very simple example of this would be
the expression of reciprocals by the scale x —-y (orX=“").
yi. X
Obviously such a scale along an axis would straighten out all
curves in which one variable varied with the reciprocal of the
other, and if both axes be plotted on such scales, then curves
would straighten out when the reciprocals of both variables
vary together.
Perhaps to the economist the most interesting application
of special scales lies in a recently discovered use of what may
be called a ‘^square-root projection’’ for certain historical data.
The speculation which leads to the use of this projection is
founded upon the analogy of familiar physical laws governing
4S2
SPECIALLY PROJECTED SCALES
483
the intensity of light, the flight of falling bodies, and the like,
in which one set of values varies (directly or inversely) as the
square of another. In the case of light, as every one knows,
its intensity varies inversely with the square of the distance
from its source and the area of the cross section of a beam of
light varies directly with the square of the distance. Falling
bodies travel in each unit of time over a distance proportional
to the square of the number of units of time they have been
falling, and their velocity therefore varies with the square of
the length of time elapsed since leaving a position of rest. The
idea suggests itself that certain economic phenomena may
closely parallel in their growth the growth of such natural
phenomena. It would be obvious, of course, that this method
of analysis could only be applied to phenomena which are free
of elements of organic growth or other factors, or in which
corrections can be made for such elements and factors.^
Other powers and roots may equally well be the basis of
the special projection. These have not, however, as yet be-
corne important to the economist; they are chiefly useful in
engineering and the natural sciences, where formulae and
equations are found of every type, and degree. The conic sec-
tions, the circle, ellipse, hyperbola and parabola are all im-
portant to the scientist, while the economist does not need
to go beyond the parabola and the hyperbola when he leaves
the straight-line. Indeed the statistician either in business
^ Ihe Gompertz curve, as it is sometimes called, to which much economic data
fits, is not unlike the curve which straightens out on a square-root projection, when
the curve has been plotted upon a logarithmic projection instead. In other words,
the square-root projection suggests itself for all economic data in which the curve,
like an ogive, seems to have a “die-away’’ approach to a maximum when plotted on
log-paper, as if reaching a saturation point.
The Gompertz curve is, however, probably nearer to the typical behavior of
economic phenomena through their initial stages, from discovery and through experi-
mentation and inyallation, to the final stage of “saturation” in which maintenance
and upkeep constitute the chief requisites. This curve has not yet been made the
subject of a special projection. Its formula is
y = or log y = log a+c^ log h, or loglog {—) log c+loglog b
indipring that a loglog y-scale with shifted zeros and an arithmetical x-scale will
straighten the curve, when the value of the constant a has been determined. For a
recent excellent discussion of this curve and the methods for determining the constants,
the reader should see Prescott, Raymond B., Law of Growth in Forecasting Demand,
m the Journal of the American Statistical Association, December, 1922, pp\ 471-479.
See also Running, Theodore R., Empirical Formula^, John Wilev & Sons, New York,
1917, pp. 29~35. ’
484
CHARTS AND GRAPHS
tm wm*i oomsei
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Fig. 387. The Four Lower Curves Fail to Straighten Out
on Logarithmic Vertical Scale.
or economics rarely works with such precise and inflexible
data as the scientist, and cannot so often attempt precise
mathematical generalization, in the shape of formulas and
equations.
SPECIALLY PROJECTEB SCALES 485
THE WORLD'S COlOiERClAL EOUIPUENI
8$tlmat«di Railiwiye, Steamships, Cables and. Telegraph
World, 1820-1919
(^gource:- 0* 5. Statistical Abetrap^J
Cftbldfl
(Thousand mils a)
Steoa Vaasels
(Thousand tons)
Tsisgraph
(Thousand mllss)
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Fig, 388. The Square-root Projection of the Vertical Scale Brings
Much Greater Regularity to the Curves of the Preceding Chart*
486
ftrojdotions of ths Siao C?ui^
(Any Boalo)
Periods
Fig. 389* The Two Ways of Straightening Out Semi-cycles of a Sine
Curve.
SPECIALLY PROJECTED SCALES' 487
It is a rule of general application that any curve can be
easily made straight if it does not undulate, that is, has
neither peaks nor valleys. The process of straightening re-
quires no more than the projection of the y-axis scale, that Is,
the scale of the independent variable, in such a way as to
make its calibrations record the values of the ordinates at
equal intervals along the A;-axis scale of the arithmetically
projected chart. A more detailed specially projected scale is
given by the following steps: 1, Divide either axis of the arith-
metically projected chart into uniform parts or intervals; 2,
From the points so obtained erect ordinates or abscissae to the
curve, and from their intersections with the curve project ab-
scissae or ordinates to the other axis, and read the values
thereon; 3 , Lay off these values along the first axis but calibrate
them with the original values. The object of this is always
the same, namely, to introduce in the altered scale of the
variable those inequalities and irregularities which will exactly
counterbalance the irregularities of the curve and smooth it
out into a straight line.
It will quickly occur to the student that such specially pro-
jected scales can even be made for some undulate or periodic
curves, to reduce them to regularity and uniformity of undula-
tions. By undulate curves we mean curves with peaks or val-
leys or both. Thus a sine curve can be laid out on such spe-
cially projected scales so that it forms a succession either of
semi-circles or of angles and straight lines. It is in such work,
however, most obvious that the peaks should be correctly
positioned at the maximum ordinates or abscissae, the valleys
at the minimum ordinates or abscissae and where cycles, are
different in amplitude or phase, that they be converted to com-
mon levels or intervals. The intricacy of this work is consid-
erable, its usefulness highly specialized and the economist,
sociologist, or business statistician will have little occasion
for it.
The subject of special projections and their use for the
analysis of curves is still In an elementary stage and little can
be dogmatically stated about it. Collections of the typical
NotE TO Fig. 389
The first requires the conversion of all amplitudes into percentages of the nodes,
and permits the use of variable phases and periods. The second requires the con-
version of all cycles to common units of or percentages thereof, and permits any
amplitude readings. ,
488
CHARTS AND GRAPHS
hyperbolic, parabolic, and other curves have been published,
intended as guides to the engineer to assist him in the recog-
nition of the nature of given curves. Such collections famil-
iarize the student with the curves of his equations, but are re-
stricted in their usefulness in the reverse process of equating
COLD STORAGE EOLDIROS OF EGGS
Average Monttily Stock* of "Cae® Egge" in Varehoueee
Dm ted States
Average of five years, 1916-1920
(Source*- Sux'vey of Current Business)
(Relative figures, 100 * average month)
tUelMi ®
Fig. 390 . Showing How Closely the Cycles of One Set of Periodic
Economic Data Approach a Sine Curve Wave.
curves. The trouble arises in the fact that slight changes
either in the scale of the chart or the constants in the equation
produce great changes in the appearance of the curve. It is
therefore impossible to prepare complete catalogs of curves
classified by their shapes.
It is probable that in the development of special scale pro-
jections which straighten or regularize the curves of certain
equations, the greatest advance in the science of curve equat-
Mur*
SPECIALLY PROJECTED SCALES
489
ing will be made. For very often these special scales are free
of the limitations of the fitted curve, neither scale alterations
nor changes of constants affecting the regularity of the curve
upon the chart with the correct special scale. This is a field
of graphics as yet little explored, its possibilities are just
opening up to us, and only time and continued usage will de-
termine what forms are valuable and the limits of their value.
It is clear, however, that the probabilities projection is not
the last invention of its kind nor yet is the square or powers
projection.
Chapter XLIV
FORMULAE FOR CURVES
While it is a great step forward in the analysis of data to
have plotted the curve and be able to visualize the behavior
of the phenomenon, yet the mathematician often seeks to
take a further step, and formulate from the curve a law for
the data, by which its behavior will be precisely described in a
single mathematical sentence. This mathematical sentence is
known as an ‘^equation.’’ And it is the object of the equation
to give us a general description of the data under all circum-
stances and times for which the equation is prepared. The
process of describing the behavior of a curve with this math-
ematical precision, is called writing an equation to the curve.
For a great many curves the description is extremely easy
to formulate. If you see a straight line on amount-of-change
paper, in which the two scales are alike and the straight line
slopes at an angle of 45° from the origin of the chart, you will
say at once that the y-values which the line passes through
are equal to the A;-values, or that the y-values are increasing
equally with the A;-values; and you would express this de-
scription mathematically in the sentence (or equation) y=Xj
that is, that the value of y for any is the same as the x itself
and whatever the A;-value is, that also will the y-value be. [Let
us vary the case a bit. If the y-scale be twice as great as the
;c-scale, that is, each unit on the y-scale equal to two units on
the :v-scale (the line still sloping at 45 degrees), it would not
take you long to determine that the formula or equation for
the straight-line curve is y=2^. Again, let us suppose that
instead of passing through the y-axis at the origin, the curve
passes through it at the value of 3. Now this adds 3 to each
value of y throughout the length of the curve and so you will
quickly write the formula as y—2x+3.
Or let us take a descending curve which on a chart with
equal scales on each axis describes a 45° downward slope from,
490
FORMULAE FOR CURVES 49 ^
0 1 2 3 4 S 6 0123458788 10 IX JS?
Figr. 391. The Linear Equation, ^=ax + c.
examples:
y^x y—2x
y=2x-\-S y=10—x
The value of c can be read at the intersection of the curve with the y-axis; in
other words c is the ordinate of the curve when x = o. The sign (+or— ) of a is
shown by the upward or downward direction of the curve; its value cafi be found
from the phrase, a — y—c, when x^l.
(Note. — The small letters immediately above each diagram in this chapter Indi-
cate the functions plotted to form the curves.)
let us say, the point of 10 on the y-axis. A little study will
show you that as the curve descends, the values of y diminish
from their original value by the amount of the corresponding
^-values. This condition can be expressed mathematically by
the sentence, y = 10 — All of these cases are simple and we can
observe that in all of them the curve forms a straight line. So
we may hazard a guess that whenever we meet a straight line
492
CHARTS AND GRAPHS
upon amount-of-change paper, we can write an equation to it
which will be a simple equation or an equation of the first
order, that is, both the unknowns, y and a:, will be found in
their first powers. The general formula for the straight line on
plain co-ordinates is sometimes written as y=ax+Cy in which
both a and c are constants for the particular line. (The values
of these constants were seen in the examples just given to have
been, for 1, 2, 2, and -1; and for ‘V,'’ 0, 0, 3, and 10.)
Remembering that the logarithmic projection substitutes
the processes of multiplication and division for the processes
of addition and subtraction, we may further generalize that a
straight line upon logarithmic paper will have the general
formula of logy =b log x+dy or log y=b log ^^-flog ^3, which is
the same as saying y And a little experimentation will
show you that every equation of this form {a and b being con-
stants) will appear as a straight line upon a logarithmic chart.
Here again we find a simple formulary relation which it is
convenient to determine. For an equation so simple as y =
ax-+c or y —ax^ is distinctly more convenient to remember and
apply than the plotted curve itself. Two familiar examples of
this are to be found in the computing of interest, the first
being the formula for simple interest plus principle and the
second for compound interest plus principle.
The special scales which have been discussed in the pre-
vious chapter are of course designed expressly to whip into
straight line formation the recalcitrant and unwilling curve,
and when they succeed, or even very nearly succeed, greatly
simplify the writing of equations. But as has been pointed
out, they can only be used with care, since some curves, or
short portions of curves, will behave similarly upon several
projections, and the approach to a straight line upon one scale
projections does not always indicate that the formula for that
scale is the best, or even a correct, formula for the curve. This
danger has already been mentioned and illustrated.
A little study of the various special scale projections will
show that they are all outgrowths of the simple linear equation
of the straight line upon uniform or arithmetically projected
scales. The object of the special projection in each case is to
so graduate the values of the scale that they absorb all the
powers of the variables, leaving to be plotted the remainder of
the equation, in which the variables occur in the first powers
only and which therefore form straight line curves on the
FORMULAE FOR CURVES
493
chart. Lipka enumerates eleven typical equations whose
curves straighten out upon the charts with the scales and we
shall briefly repeat this list, that the student who has found
a combination of scales which makes his curve straight may
quickly find the equation best describing his data.^
For the straight line upon uniform (that is, arithmetically
projected) scales, we have an equation in the first degree,
called, from its form, the linear equation. Its type is y ^ax-^Cy
in which and c are constants whose values can be easily found,
c being the intersect point of the curve upon the ^^-axis and a
being the tangent of the angle of the curve upon the x-zxisy
easily computed from any observation after c is known. For
all curves which pass through the origin, the equation is re-
duced to y—axy since c has disappeared.
For the straight line upon log paper, both scales being
logarithmically projected, we have, as we have seen, the equa-
tion log y—h log X’^dy which is but another way of saying
y—ax^ in which Uy b, and d are constants, d being the logar-
ithm of dy and both d and b are as easily found as c and a above.
Drawn upon arithmetical paper, the curve is, of course, not a
straight line, but becomes a simple parabola or hyperbola.
It is a hyperbola, that is, in this case, a falling curve, if b is
negative; if b is positive, the curve is a parabola, that is, in
this case, a rising curve, and approaches the vertical as it in-
creases if b is greater than unity and approaches the horizontal
if b is less than unity. In the one case where b is unity, the
curve straightens out (on arithmetical paper) since here b can
be omitted from the equation and the latter becomes y==ax.
Thus we see that the equation y — ax will be a straight line upon
either the arithmetical or the logarithmic projections. This
is the same as saying that if a straight line curve on arith-
metical paper pass through the origin, it will also be a straight
line upon logarithmic paper.
We have in the previous chapter mentioned the shifting of
the zero point upon a logarithmic projection, that is, the re-
calibration of the scale after it has been graduated (plotted.)
^ The remainder of this chapter is largely and very inadequately drawn from Pro-
fessor Lipka^s excellent book, Graphical and Mechanical Computation, John Wiley &
Sons, 1918. This has been done not to substitute the present volume in any way
for that treatise; it has rather been the writer’s purpose to draw attention to the
extraordinary possibilities opened up by Lipka’s work, and to direct readers to it.
The volume is indispensable to the student, and to the technician it will almost certainly
open up a new world of research, arming him with invaluable implements.
.7 ,8 .9 1 . 2 3 4 5 8 7 8 9 10 18
Fig* 392. The Curve of y^ax^ or log y=Iog log x*
See footnote on opposite page.
FORMULAE FOR CURFES
495
The scale then represents, of course, log {y -c) where c is the
constant which has been added to the plotted values to give
the calibrated scale-figures. When such shifting of the scale
has straightened out a curve upon logarithmic paper, the curve
has, of course, the equation log (y -c) —b log x+d (instead of
the immediately foregoing log y = b log x+d). From this new
equation we derive y -c =ax^ and so y =ax^+c. Thus we see
that the shifting of zero-points on the log scale is but an ad-
justment which makes c disappear and gives the straight line
on log paper. On log paper without shifted zeros, the curve is
parabolic, concave to the x-zxis if c is positive, convex if it is
negative.2 When c is zero it disappears from the equation and
the latter becomes y=ax\ an equation already described,
having the straight line form on unshifted scales of log paper.
And when b becomes unity it disappears from the equation
leaving an equation of the first type, forming a straight line
on arithmetical paper.
The three types of equations are closely related, all having
the general form y=ax^+c^ in which b is unity for the first
type and any number for the others and c is zero for the second
type and any number for the others. The third is distinct
from the first and second in that it contains not one or two,
but three constants, and hence requires calculation (when we
are seeking to find the proper shifted scale) for the third con-
2 The value of c can be computed by taking any three items in the data (or points
along the curve plotted experimentally on uniform paper) such that their ^;-values
form a geometric series, i.e,, xi:x 2 ::x 2 :xz. Then
Xi = \/xiXs
and
ax^b^'s/ axi^ axj^
Hence
y2—c== V (yi—c) iyz—c)
and
yiys— y2®
c —
yi+ys— 2y2
So we must observe the ordinates, yj, y 2 , yz at these points and substitute them in
the last equation to get the value of c.
SCALE PROJECTIONS OF FIG. 392
Arithmetical
Logarithmic
The value of a is shown by the ordinate of the curve when x=l (i.e., log x—0).
The curve is hyperbolic when b is negative and parabolic when it is positive.
(When ^—0, the curve is a straight line parallel to the .v-axis. If 1, the curve
is also straight upon the arithmetical projection, its equation, y^ax being linear
(see the curve y—x). The value of b can be found from the phrase, ^ = log y—
log ay when (i.e., log a; = 1).
496
CHARTS AND GRAPHS
-1. a - - — ■ — -
1 £ 3 4S6789 10 1 2 3 456789 10
Fig. 393. The Curve of or, log (y-“c) = log a+fe log x*
examples:
SCALE projections:
Arithmetical Logarithmic
Logarithmic with shifted zeros
The scales of log {y-c) in the two lower diagrams are logarithmic projections with
the scale-figures altered by the value of c; when c is negative the scale of log
(y-c) will include the value of zero. The value of c must be known before the
scale can be so altered. Four curves such that they straighten out on these
shifted scales, are shown, two curves for each scale, one ascending and the other
descending. These four curves are also shown on other projections, showing
their various shapes. The value of a can be found by the phrase, c when
Ar==l (i.e,, log ?c=0). The value of b can be found from the phrase, ^=log (y— c)
—log a when .v^lO (i.e., log — 1).
FORMULAE FOR CURVES
497
stant, c. For the third constant is not readily capable of
graphic solution, save on arithmetic paper; it must always be
known or mathematically calculated before a proper altered
scale can be found. Of course when we have by experiment
found a satisfactory scale it amounts to a trial and error
method of solution.
We also noted in the previous chapter the projection of
powers of variables along the scale, the use of the square-root
projection being illustrated. In these scales the values of x
have been entered as calibrations or scale-figures at points
which were plotted or graduated for the values of being
the known exponent of the power. A curve therefore which
straightens out upon this when the y-axis is uniform (arith-
metical), has the formula, y — ax^+c, in which the constants,
a and r, are found as before. This equation is of the same
general type as the foregoing, y=ax^+Cy its only difference
being that J is a known, not an unknown constant and is
called k therefore. Here b or k plays the role of the third
constant, being known. And when ^ is a small positive in-
tegral, such as 2, this scale projection aifFords a simple means
of straightening the curve, but unlike r, b cannot be easily
calculated when it is unknown. The method therefor is lim-
ited to the use of curves in which b is known, and is valuable
in such cases when ^ is a small integral. It is much easier, for
example, to prepare a squares projection than to shift the zero-
point on a log scale.
The squares projection is an example of the powers projec-
tion in which the exponent of the power is a positive integer.
If on the other hand, the exponent be fractional, we have an
inverse power or root. An example of this would be a square-
root projection. Or the exponent may be negative. The
simplest instance of this is the reciprocal projection, for the
reciprocal of a number is its -1 power. When a curve straight-
ens out on a chart one axis of which is reciprocally projected
(the other arithmetically) its formula is obviously y=-^ —
a modification of the general type formula y =ax^+c (in which
k = -1) or y =^ax^+c (in which b = -1). Every powers pro-
jection, therefore, when used along one axis only, always
straightens out a curve whose formula involves that partic-
ular power of the one variable. We may treat all the possible
FORMULAE FOR CURVES
499
powers projections as but one class, with formulas of the
type y ^ax^+c. These formulas contain three constants, only
two of which, however, are unknown. The third constant is
known, and is the exponent of the variable. And surely it is
clear that whenever the power of a variable is known, that
power may be laid off upon the scale for that variable so that
the plotting of only the first power of the variable (that is,
the variable itself) thereon, will make the curve a simple linear
one. The power remains in the scale and hence in the formula,
but has vanished from the curve.
In all powers projections so far considered, we have used
the special projection upon one scale only, the other being
uniform. The equation being y^ax^+Cy it is clear that the
;c:-scale has been specially projected, for the given power, ky of
the variable, x. Care must be taken to keep this arrangement,
for a reverse arrangement will fail to straighten out the curve.
Only in the case when c = 0 and the equation reduced to y = ax^y
is it immaterial which scale be subjected to the powers projec-
k —
tion, for here we may write y =ax‘- or Vy =a'x, in which a! «
h
-A a. The line will therefore be straight either upon the
powers projection of one scale or the corresponding root pro-
jection of the other, y—ax^ is, however, too easily straight-
ened out by the log projections, as we have seen, and hence
the case is of no value. The real use for the powers (and
roots) projection of the y-scale is in the wholly different equa-
tions of the form y^=^ax-^c (including
Closely related to the projection of reciprocals, is the pro-
jection of products. Thus the equation y =-4-^ may be writ-
ten xy ^ a ’^■cx. In this form we see that the equation will not
only straighten out upon semi-reciprocal scales, y, but also
SCALE PROJECIIONS FOR FIG. 394 :
Sfluare of X, Square of Y.
Arithmetical
Square root of X. Square root of 5'”.
Reciprocal of X. Reciprocal of Y,
Six typical curves are shown by full lines, one on each of the specially projected
scales and all upon the arithmetical projection. They straighten out only upon
the scales on which they are plotted. If, however, c=0, that is, there is no added
constant and the equation is reduced to y = ax\ then the curves are straight
upon either of two different projections, thus on y or on at, or
y; and on 1/a, y or x, \fy. Such curves are shown by broken lines.
Fi^* 305. The Hyperbolic Curves, y= — -j-c and ya= 5 —
X a-j-cx
See footnote on opposite page.
FORMULAE FOR CURVES,
501
upon the semi-product scales, Xy xy- Product and quotient
scales are, however, a little hazardous, in that the introduction
of one variable into both scales may often force a curve to ap-
proach nearer to, though not entirely to, a straight line, with-
out the least real significance. Moreover, they require some
computing, as the series of y original data must be replaced
by the series xy^ in which each value of is multiplied into its
corresponding value of y. To be sure, no special scale need be
projected, the products or quotients being put upon a uniform
(arithmetically projected) scale.
1 The most interesting use of the reciprocal projection is for
the equation in which both variables are in reciprocal form,
namely y “^+c. This is the equation of the ordinary
hyperbola. On uniform scales, it is asymptote to the co-
examples AND SCALE PROJECTIONS FOR FIG. 395:
Single Reciprocal Produciy xy
Double Reciprocal
i=l+l
y 3 a* 3
Quotient, x/y
X 2.x
xy = 3
Arithmetical
Both
Quotient, y/x
Two typical equations of hyperbolic curves are shown here. The one shown by
the broken line, involves the reciprocal of one variable only and straightens
upon a single reciprocal projection; from it a series of products of the two vari-
ables can be computed which’ straightens upon arithmetical paper on the plotting
of X and xy. Note that its asymptotes are 0 and c; and that the value of c can be
found by inspection at the intersection of the curve with the y-axis, calibrated
as infinity, on the reciprocal projection. The value of a can be found from the
phrase, a-y—c when a = 1.
The other equation, shown by the full line, involves reciprocals of both variables
and straightens out upon the double reciprocal projection, from it two quotient
series can be computed which yield straight lines. Note that its asymptotes are
y^l/c and x — —alc and can be read at the intersections of the curve with the
axes, calibrated infinity, upon the reciprocal paper; from which the values of c
and a can be easily found.
(Note. — ^The small letters over each diagram in this chapter show the functions
(of the variables) plotted. Large letters, X and Y, indicate the scale-figures or
calibrations, and are omitted if these are graduated for the same functions; the
curve can then be plotted directly from the scales without finding the functions.
The presence of the large letters indicates that the curve cannot be directly
plotted from the scales, that the indicated functions must first be found and these
(instead of the variables) must be plotted from the scale-figures.)
502
CHARTS AND GRAPHS
Fig. 396. The Hyperbola with Three Constants, y—
SCALE projections:
Arithmetical, Double reciprocal, one shifted.
Quotients, singly shifted. Quotient, doubly shifted.
A single typical equation is shown in hyperbolic form on arithmetical scales and
straightened upon reciprocal scales, one of which has a shifted zero. The re-
ciprocal scale with the shifted zero can be prepared only when the added con-
stant, d, is known, since the zero is shifted by its amount. Also when d is known
the quotient series, — or — , can be computed from the data and will
yield straight lines upon the plot of either, x, \
y—d
jOT-—, y.
When d is un-
FORMULAE FOR CURVES
503
ordinates a; =
a j I
= «= and y ,
c c
1
we can see
From its form, - =
. y
that it will straighten out upon paper in which both scales
are reciprocally projected,"”,—. If for any reason we desire a
' pc y
chart giving detail to different parts of the curve, we can re-
X
write the equation as - = a+cx and then we see that by the
use of the scales, in which both are uniform and one, the
y .
y-scale, is used for the plotting of the quotients of the values
of X by their corresponding y-values, the curve can again be
straightened out. Or we can cast the same equation into the
\ a ay
form“"-i:= — or 1 == — and so see that the curve will
y
straighten out upon the scales, y, Here are three arrange-
X
ments by which the ordinary hyperbola can be straightened
out. Take your choice. The constants and ‘V can be
found by inspection from the plotted curve on the double re-
ciprocal paper.
X
If we write this equation in its usual form, y , it
a “j— cx
will occur to the student that it may occur in modified form with
X
a third added constant, thus y = — ; — -^-d. The curve of
this modified equation will still be hyperbolic upon uniform
paper, but will no longer pass through the origin of the chart.
Just as in the first and third equations discussed, so here the
added constant may be determined by the ordinate of the
curve at the zero-point on the ;c-axis, that is, the point where
known, the zero must be shifted along both axes to some known point in the
curve, and if we call the co-ordinates of this known point (it may be any we wish
to select) xq and yoj then we can compute the quotient series ^ ^ which
yields a straight line y upon the plot of a:,
y-yo*
The value of d is obviously
shown by the intersection of the curve with the origin, that is, when
Note that the asymptotes, y — l/c-\-dy and x~—aU^ are shown upon the recip-
rocal projection by the intersections of the curve with the axes, calibrated infinity;
from this the values of r and a are easily found.
504
CHARTS AND GRAPHS
the curve intersects the y-axis upon arithmetically projected
scales. This cannot be read upon the reciprocal scales as the
latter never reach zero, since zero \vould have to be plotted
at its reciprocal, infinity, a manifest impossibility. We must
therefore first plot the curve on uniform scales to determine, if
we can, the value of d by inspection. Then if we write the
X \ a
equation in the form y —d — —r — and then j = - + c^ we
shall see that the curve will straighten out upon the reciprocal
scales
1 1
y
In short we have now come to the use of
shifted or false zeros upon the reciprocal scale. The curve can
also be straightened out by the quotient scales, x , —
y-d
and
X
be done.
In every case there is a great deal of computing to
When the value of d is not easily found, it may be advis-
able to use a method of differences, which has not so far been
mentioned. Virtually this amounts to shifting the zero-point
(or origin of measurements of the co-ordinates) to any con-
venient point we wish along the curve. To do this we first
select a point upon the curve or an item in the co-ordinates
which we may indicate by and yo. Then we compute the
difference between these co-ordinates and all other co-ordi-
nates, X and y, in the data. Finally we divide the differences
to get the series of quotients ^ ^ and plot these as co-ordi-
y -yo
nates over the abscissae of a;. The reason for this is that the
equation, y= [-J, can be reduced, by subtracting the
a+cx
selected point, yo= \-dy to the form ^ = a + cxo + cx
^3 ^+cxo y -yo
-1 XqX, The added constant, which has caused all the
a
trouble, has been, as you see, eliminated, and the constants,
Uy c, and ;Vo, are left. Write the second half of the equation
c •
as (a-l-c:vo)+- {cL+cx^ Xy and you will see that the curve will
(Z
straighten out on the plot of
y --ya*
Of course this is not
FORMULAE FOR CURVES
505
a plot of the original series, it is only a plot of the cotangents'"^
of the points upon the curve from the selected point in the
curve, but if the derived curve be straight it is proof that the
original curve has the formula y —■ — \-d.
a+cx
In all reciprocal scales the constants a and c are easily found
by inspection. The axes of the scales are always calibrated
as infinity along each scale and cross the curve at its asymp-
totes. The asymptotes have the value x-- and (y ~d) = - .
c c
So we need only substitute the observed values for a: and y to
obtain the constants and complete the formula.
Similar methods can be used to straighten out the ordinary
parabola, the equation for which is y —a-\-bx+cx'^. (The
reader will note that to arrange the variables in ascending
order, we have altered the symbols for the constants, and
that this equation is really a modification of the simple linear
one, y-ax+c.) A little study will show that the value of
the constant a is the value of the curve at its intersection, ex-
perimentally drawn, with the y-axis. Since the equation can
be written — t^'bJ^cXy it is obvious that if we know a. we
can compute the series of quotients and that the curve
of these will straighten out upon the scales, .t, ^ If we do
not know the value of we can use the method of shifting
the origin to a point upon the actual curved and will get a
straight line by plotting (not — ^ because recip-
X —Xq y -yo
rocals are not involved in the equation). If the values of a;
in our data form an arithmetical series, we can take the suc-
cessive diflFerences of the y-values, that is, A y, and will find
that the plot of A y, is a straight line.^ The polynomial in-
volving higher powers sych as y =^a-\-bx+cx^+dx^+ . . . must
be successively differentiated or the method of determinants
^ Cotangents only because they are reciprocals.
4 By substituting yQ = a-\-bxQ-\-cxQ^ we get y-yQ=^h{x—x,i)^c{x’^-~x\) or
y-~yo
— - - ^h-\-cxQ'\-cXi in which the phrase is a constant.
X — AO
s The formula for A y is to be found in Lipka, '^Graphical and Mechanical Com-
puiationF P- 146.
5o6
CHARTS AND GRAPHS
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Figr. 397. The Parabola, y = a+6x4"cx®.
See footnote on opposite page.
FORMULAE FOR CURVES 5^7
used, but the mathematics in such work is outside the scope of
this volume. Indeed, the phrase cx^ may be added to any of
the foregoing equations and will produce the same difficult
results.
The close observer may note that so far no mention of the
semi-logarithmic chart has been made. The curve which
straightens out upon it is known as the simple exponential or
logarithmic curve. The formula of such a curve is of a wholly
different nature from any so far considered, for in the latter
all exponents have been constants. In the equations of expo-
nential curves, we meet with variable exponents. The simple
exponential curve has the formula, y—ab'^. The curve upon
arithmetically projected scales is convex to the ;i:-axis, rising
rapidly as increases positively, and having a horizontal
asymptote to the x- 2 ixis as a; increases negatively and inter-
secting the y-axis, at the value of since becomes unity
at the y-axis {x being zero there and the zero power of any
number being one). Now if then of course log y =
log +a; log ^ = (in which a' and F are constants, the
logarithms of a and h). The second part of the equation be-
comes familiar enough in this last form, being the very first
type considered. So if we plot the curve upon semi-logarith-
mic paper, a:, log y, it will straighten out. And conversely all
straight lines upon semi-log paper have the equation y^aF.
And this formula applies to all historical data which have
straight-line curves upon semi-log paper. It is the formula
of the law of organic growth.
Several variations of the exponential curve equation are
obviously possible. If a constant be added we havey —ab^+c.
This we turn into y -c = aF and from it we derive log (y -c) =
log a+x log b or log (y — r) +b'x. Here we see a need for
the shifted zero upon the logarithmically projected scale, an
the curve straightens out upon the scales a:, log (y If
c is unknown it can be computed from the experimental curve
I^SCALE PROJECTIONS FOR Ffc. 397:
Quotients {shifted) and differences.
Arithmetical.
The ordinary parabola (shown by a full line on the lower diagram) is the sum of
three different curves (shown by broken lines) and cannot be straightened out
upon any useful projection. It can be made to yield, however, various straight
lines for series which have been computed from its known data and these afford
a test of its equation.
5o8
CHARTS AND GRAPHS
examples:
y—ab^i or log y— log a~{~x log b
y — ab^'\'Ct or log (y — c)=log a-\-x log b
y — ab^c^^y or {l/x) log (y/fl)=log a-^-x log h
The simple exponential equation y — ab^ straightens out upon semi-log paper.
When a constant c is added, the zero must be shifted by the amount of the
constant on the log axis; one example of this (shown by the broken line), is plotted
upon three diagrams to show its behavior, another which on arithmetical paper
parallels the simple equation shown, is plotted on two (shown by the dot-and-
dash line). These curves are hyperbolic. When a higher power of the variable
is added, the curve becomes parabolic and cannot be straightened. A quotient
series with shifted zeros can, however, be computed, namely, >
FORMULAE FOR CURVES
509
upon uniform paper.® Another exponential curve is that which
has the formula y =aUc^. If it be written as log y = a' -\-b'x +
c'x^ (in which the primes of the constants again represent their
logs) we see that it is very similar to the ordinary parabola,
and indeed, precisely the same methods must be used to
straighten it. If a is known, we can plot it upon the uniform
scales X, (
log y -log
X
If a is unknown we can use either
of the other two methods, and derive series whose curves
straighten out upon the scales x, (
!g g ._ ?Lj 2 g j!) and
A log y. Other exponential curves have still other formulae,
which are often but modifications of any of the foregoing
through addition of other variable powers, such as df" in the
equation y ==a+hx+cd''. These more complicated equations
must be subjected to even more devious calculations before
derived series can be found which straighten out and prove
the equation.
The reader should not consider from* this brief summary of
the scale projections which straighten out non-periodic curves,
that all or even nearly all curves can be straightened out by
them. And the non-mathematical reader will doubtless have
a wholesome respect for the processes of curve equating even
by the above methods. He will probably find little difficulty
with the simple linear, the simple parabolic and hyperbolic,
and the simple exponential curves, requiring as these do only
the arithmetical, logarithmic and semi-log charts. But some
curves are immensely difficult to exp.ress in equation form, and
must often be broken into parts with separate equations for
each part. It is true that these parts can be collected with
proper mathematical symbols of limits, into a single equation
and in this sense it is true that an equation can be written to
any curve in the world.
But the long and complicated equation has little value.
The equation for very irregular curves — such as the profile of
a man’s face — may take up more space than the curve itself.
The disadvantages of complicated formulae are many. For
one thing, a very complicated formula is difficult to understand
® The same method for finding c can be used as before^ for the equation y = ax^ c.
510
CHARTS AND GRAPHS
even when it has been stated — the average person still has to
plot its curve to understand its meaning. For another thing,
very complicated formulae suffer from the danger of being
made unnecessarily detailed or intricate by chance variations
in the observations which form the data.
This last consideration, the danger of chance variations in
the observed data, leads us to the thought that the “true
curve’’ for the data, if all errors were absent, might be a very
simple curve, easily expressed by an equation, while the curve
of the actually observed data remains irregular and compli-
cated. We therefore oftentimes have to be satisfied by simple
curves which closely approximate the actual curves, when
such simple curves can be found. And the problem then
becomes one’ of “fitting curves’^ with the best possible (that
is, the closest fitting) straight lines, in the attempt to find
simple and approximate descriptions and equations.
The reader will have seen by this time that much of the
care expended on proper curve plotting has for its purpose the
clear visualizing of the phenomena, but that still other care is
expended in the attempt to capture the curve in a symmetrical
or regular formation. And he will now see that one of the chief
purposes of symmetry and regularity is to enable us to formu-
late laws governing the behavior of the phenomena repre-
sented by our data and curve. In the discussion of fitted
straight lines, which is so far as it seems desirable to enter
the subject in this book, he will be reminded of the “trend”
and “secular change” discussed previously in historical curves;
in fact for historical series the secular trend is often con-
sidered to be a fitted straight line. And he will now also see
that these secular trends can be expressed mathematically in
equations. He will also see that the operations of interpolation
and extrapolation can be even more precisely performed when
the equations are used than with charts only. He will see, in
short, that the possibilities of mathematical description or
summarization of curves opens up to him a valuable adjunct
to the use of the curves themselves.
PART V.
CALCULATING CHARTS
Chapter XLV
CURVES FOR FORMULAE
Having seen something of the way in which formulae or
equations can be written to curves, we can reverse the process
and prepare curves to illustrate formulae. In this way, we no
longer seek the mathematical statements describing a curve,
but we seek the curves illustrating a mathematical statement.
The advantage of writing an equation to a curve lay in the
fact that, from the equation alone, we could, by mathematical
operations, find the values represented by each or all of the
plotted points along the curve; the advantage of drawing
the curve to illustrate an equation lies in the fact that without
bothering about the mathematical processes, we can read the
values represented by the equation directly at a glance from
the chart. In short, the chart may be made a substitute for the
processes of calculation and computation, and the chart then
becomes a calculating machine.
If, as in the previous chapter, we have a curve for which
the mathematical equation is T = 2 X+3, and we wish to find
the value of F, when X, let us say, is 5, we do not have to
solve the equation by mathematical processes, multiplying
5 by 2 and adding 3, but from a glance at the chart we can- see
the F-value of that point on the curve whose A’^-value is 5.
We follow the ordinate from 5 on the ^r-scale up to the curve
and from the intersect point (where the curve passes through
or intersects the ordinate) we follow the abscissa or horizontal
CO the y-scale and read 13, the answer. In this case, it is true
that the mathematical operation of solving the equation seems
simpler than the graphic one for the reason that we have
selected for illustration of the principle a simple mathematical
equation. ^ But you will find many complicated formulae and
equations in which the mathematical operations are far more
tedious and lengthy than the graphic process. In such cases
it will be useful for you to be able to construct calculating
512
CHARTS AND GRAPHS
curves and charts by which mathematical equations of the
given type can be readily solved.
The purchasing agent, perhaps, buys in foreign markets
and must multiply his quotations by the prevailing rate of
foreign exchange and add perhaps certain local charges in
this country, before he can compare the values of different
offers. To interrupt telephone conversations with these
mathematical operations would perhaps be difficult, but he
could be provided with a special chart on which he would see
at a glance the real value of offers without interrupting his
telephone conversation to the parties concerned,
T = 2X + S
That a single straight line curve upon an arithmetically
projected chart-field will illustrate a simple mathematical
equation involving only two variables in the first degree, we
already know, for any straight line upon arithmetically pro-
jected chart-fields has an equation of the general form Y=
aX -|-c^(in the right side of which a and c are given constants
and X alone is variable). We can, however, by a series of
such straight lines show the equation for two independent
variables. Let us assume for example that c is a variable and
call it Z and that the constant <2 is 2. In other words let us
prepare a calculating chart for the equation 7=2 X+Z. As
we have seen in the last chapter the figure 2 determines the
CURVES FOR FORMULAE
513
slope of the straight line curve and if the ;v-scale is only half
as great as the 31 -scale then the slope of the straight line would
be rigidly 45° to the ;v-axis of the chart. The added element
2 » T - 21
Fig. 400.
Z merely determines the height or position of the straight line
curve upon the chart; the straight line curve passes through
the origin of the chart when Z is 0 and in general intersects
the 3 ;-axis at the value of Z because at the y-axis the value of
X is 0 and the equation is T=Z. Now because Z itself is a
variable we cannot show the equation by a single straight
line but must use a series of straight lines, each for different
values of Z and must therefore mark off a scale of Z upon the
straight line curves themselves. The result is a chart with a
series of parallel straight line curves which are diagonal upon
the chart and enable us at once to find the values of Y when
y'=2 X-hZ. To read a certain value, as, for example, when X
is 5 and Z is 3, we need merely read up the ordinate from the
point 5, on the .r-scale, to the diagonal line or curve marked 3
on the z-scale, and from the intersect of this particular curve
with the ordinate, read horizontally across the abscissa to the
point on the y-axis where we find 13, the answer.
CHARTS AND GRAPHS
SH
To use this chart for subtraction is very easy, for we
merely reverse the process and the dependence of the vari-
ables, saying that if F = 2 X-[-Z, then Z = Y —2 X. If T = 10
and X = 2, then we read across the abscissa from the point of
10 on the y-scale to the ordinate from the point 2 on the
^c-scale, and note the value of the diagonal which passes
through this point, namely 6 on the z-scale. It is of course
not necessary to use whole numbers either upon the chart or
in the equation for we can easily interpolate between the
2
X
2 » 2X t Y
Fig. 401.
actual ruling on the chart to estimate, very closely, the values
desired, when they are fractional. These subtractive charts
can also be made to show the difference, not upon the straight
line diagonal curves of Z, but upon the y-axis itself, by making
the curves express the general equation Y^Z -aX and making
the diagonal curve a descending instead of an ascending one,
as illustrated in the previous chapter. Still another method
for obtaining the same result would be to carry the chart de-
scribed in the last paragraph down into the negative side of
the A?-axis.
CURVES FOR FORMULAE
515
Indeed the calculating chart only becomes difficult to
understand when we begin to talk about it. The simple chart
is much more easily made than described. Yet it is necessary
z
2 = Tf - 2X
Fig. 402-
for charts of the more complicated formulae, that the elements
which go to make up the simple chart be clearly defined. And
the first consideration of importance is the distinction between
physical distances upon the chart and numerical values
assigned thereto. If we call the two axes of the chart (;c), and
(y) and the diagonal dimension (x), then we have at least
distinguished three different possible places in which scales
may be projected physically and given numerical values. If
we indicate the physical distances along these scales, measured
from an origin-point, as a;, y, and s, respectively, and the
numerical values finally assigned to these distances (i.e., the
scale figures) as X, T, and Z, we have a simple means of
keeping two more details separate in our minds.
The importance of distinguishing in this way between
final calibrated values or scale-figures, X, T, and Z, and the
5i6
CHARTS AND GRAPHS
actual plotted scale-distances x, y, and z, cannot be under-
estimated, for confusion at this point will baffle the student
for the remainder of his work upon calculating charts. It
is to be understood that the small letters, x, y, and z, are
merely essentials in the planning and making of charts; they
do not appear upon the finished work. It is to be understood
that the large letters, X, Y and Z, are merely symbols for the
different variables in the equation to be calculated. If these
variables are indicated by other symbols in the equation, then
the large letters will not appear on the finished work, but the
accustomed symbols will be substituted for them. The large
letters are useful in planning the work as they clearly indicated
the axis or scale upon which the variables will appear. If no
better symbols are to be had, then the large letters and
Z, one or all, may be retained upon the final chart and its
formula. Indeed, even the small letters, x, y, or z, may be
finally used for this purpose; they then of course indicate the
variables and scale-figures and have no application to scale-
distances. But during the stage of making the chart, we shall
always use the symbols consistently in the meanings specified.
Thus if we have the equation “income - operating expense =
operating profit” or “i -e =p,” we shall substitute Y, let us
say, for i; Z for e; and X for p; and write Y -Z=X. When the
chart is finished, we shall substitute the original symbols
again, and write i -e—p.
But between the scale distances, x, y, and z, and the final
calibrations, X, Y, and Z, an elaborate structure of modifica-
tions and substitutions may be built up. These are necessary
for very complicated formulae; in simple equations they fall
together like a house of cards and can be wholly disregarded.
Thus if our equation be Y = Z-{-Z, we can obviously lay off
the distances, x, y, and z, directly from the equation. We may
even use the same face of the ruler for both y and x, that is,
along the axes of the chart, plotting the diagonals to conform
to the equation. But when Y-2X+Z, the chart becomes
very tall, and as we have seen, it is just as well to lay off the
Y and X scales differently.
Since we have occasionally in this way to use different
units of measurement in laying off scales, it is well to have
clearly in mind one common unit of measurement for the entire
chart. This unit we call the “modulus” of the chart; and it
does not matter whether the modulus be one inch, one foot,
CURFES FOR FORMULAE
517
one centimeter, or any fraction of these, so long as it be the
same for all parts of the chart the proportions of the various
parts of the chart are the same. The modulus then is simply
a general unit of distance which serves in planning the chart
to equate the scale distances and^the^scale values; thus, ^ = wX,
or x^lmX.
Now it is a great convenience to plot distances directly from
the data, that is, the values or scale-figures to be assigned.
When this can be done we can copy scale-figures directly from
our ruler as we plot. And here secondary moduli for each scale
become useful. These’^are merely fractions or multiples of the
chart-modulus, and when they differ from the latter, may be
indicated by or Thus when x~mXy but
when x=2mXy mx—2m. In the charts already considered, we
have seen the chart of Y = 2X+Z made with the horizontal
units of measurement twice as long as the vertical ones. If
the vertical units be w, then the horizontal ones are 2w.
Experience will show that it is best to proportion a chart
in such a way that all intersections be as sharply drawn as
possible. The object is to make readings from the chart
accurate. If two lines are perpendicular, there can be little
doubt about their intersection point, but when they cross at
small angles (that is, are nearly parallel) it is not so easy to
decide the exact point of intersection. Since the and y co-
ordinates are perpendicular, obviously the s-diagonals cannot
cut both co-ordinates more sharply than at 45 degrees. So
the most desirable form of chart is one in which the s-diagonals
form about 45^ angles with the axes. And it is the primary
purpose of the scale-moduli (not the chart-modulus) to pro-
duce this condition. When Y — X + Z and -jc and y have equal
moduli, the z-diagonals, as we know, have the right slope.
And so when Y -2X^7. it is easy to see that the modulus
of the AT-scale (letting viy — vi) must be before the
diagonals will have the same slope. Here we may note that
irix
•^ = 2, the coeflicient of X in the equation. And it is a use-
ful empirical rule that the coefficients of the variables (on the
axes, that is. A" or F) are the ratios of their scale moduli to
the chart modulus.
The scale moduli (as distinct from the chart modulus)
serve still another purpose, for since they form what we might
5i8
CHARTS AND GRAPHS
call “plotting instructions,” they can be used to indicate the
side of the engineer’s hexagonal rulei which is to be used. Thus
if we use a chart modulus of one inch, we can plot m from the
10-side of the rule, fm, from the 20-side of the rule, |m from
the 30-side, and so on.
From the outset in chart making for formulae, we must
keep in mind the desirable limits of the variables to be shown
by the scale figures. If our chart is to be used in calculating
a few pounds, it would be foolish to make it include tons as
well, for then the scale for pounds would be so small that it could
not be accurately read. If the price of paper is quoted in
cents, why make a chart which shows millions of dollars, and
on the scale of which cents are so small as to be invisible.
Obviously the larger our scale becomes the more clearly it
can be read, and the more accurate will be its calculations.
Hence we should try to include in the range of the scale only
its useful parts that we may make them as large as possible.
This calls for the setting of limits f r the range, an entirely
arbitrary matter, for which it is only necessary that we know
the extreme high and low values of the variables which will
be met with in the use to which the chart will be put. Having
determined these values of the independent variables, we can
write them into our formula by a convenient trick, thus
10 , 12 j , . r ,1 22 10 12
Q +2 2 and hence, in full y ^ =x q +2 ^ •
Now we know how much space to give to the chart, or how
large to make the chart-modulus for a chart of a given total
size.
We are now in a position to consider the havoc wrought by
constants in a given formula for which we are making a chart.
If these constants be coefficients of the variables, we have an
equation of the type hY = aX-\-cZ, we shall have the scale
moduli, OTj, = hm, and = am. The scale modulus of the z-scale
for diagonals need not be calculated, Z is much more easily
entered upon the chart from observations of the actual values
for various points after the co-ordinates have been calibrated.
The formula bY = aX-\-cZ can be written Y^-^X+^Z:
0 0
this will enable us to make Wy^m and = which may
‘ For a description of the engineers’ rules or scales, see Chapter XV'II.
CURVES FOR FORMULAE
519
give an easier plotting scale directly from the ruler. Thus if
we have 14T = 7X+3Z, it is a convenience to plot y—mY and
7
x=-Y^inX = ^mX, for we can plot and calibrate directly from
the 10 and 20 sides of the ruler; but if we have SY = 2X+3Z,
y X
or it IS more convenient to plot y — and
x = \mXy for we then use the 20 and 50 sides directly.
Of course these considerations are largely directed at the
simple co-efficients, but they hold also for more complicated
ones. When we have an equation such as 157 =37.29SX-fZ,
no rulers will serve directly and it would not pay us to plot
37.295 . ,
— mX = ,237 mX from a specially constructed scale
(best secured by the method oftriangulation^), instead we need
only plot x^.lSmXy which we can do from the 40-side of the
rule, and shift the direction of the ^-diagonals a little. When
constants are added in the equation, the effect is not to enlarge
or diminish the size of scales, nor to alter the scale-moduli in
the least, but it is to shift the scale numbers, without otherwise
disturbing them, along the axis. No matter how many con-
stants be added, they can of course be lumped into one, thus
bY — aX+cZ+k. Obviously the correction for the constant
must be made upon one or another scales, that is the constant
must be attached to one or another variable, to get it into the
chart. Thus {bY--k)^ aX+cZy bY ^{aX+k)+cZy and bY^
aX+{cZ+k) are all forms of the same equation. The amount
of shift is proportional to the amount of ky but care must be
taken to divide it by the coefficient, if there be any, of the
k
variable to which it is attached. Since aX’\-k—a{X +‘ — ), we
k , , . ^
must make x=ma(X+—) and if we wish to shift the scale
(i.e. add the constant) after the scale (i.e. x=amX) has been
plotted we must shift it by the amount of , not of k. It is
a
sometimes simpler to make the correction while plotting, that
k
is plot for x-in{dX+k) or x^am{X+—)y directly by sliding
2 For a description of the amplifying or diminishing of scales by triangulation, see
Chapter XVI L
520 CHARTS AND GRAPHS
the ruler along until the calibration X is at the point of
We have so far considered only one form of calculating chart
the distinct feature of which is the parallel straight line z-
diagonals. This is the chart for all equations involving the
sum or difference of two variables of the first degree. It may
be called, therefore, the additive chart. It is by far the most
important, and useful, as well as the simplest chart of its kind.
Moreover, it is the basis for so many other calculating charts
that we have dealt with it in great detail, almost all of which
will be essential to an understanding of the other types of
calculating charts.
The reader may note, however, that the use of the scale-
modulus for the projection of the z-scale has been expressly
enjoined. This is a peculiarity of the rectangular chart, the
z-scale being best laid off by inspection in it. The reason for
this is that the scale of the z-diagonals is not easily commen-
surable with the X and y^scales. The diagonals form angles of
45 degrees with the other co-ordinates, when the scale-moduli
of the a: and y-scales are precisely adjusted; they form approxi-
mately the same angles when the adjustment is not complete
CURVES FOR FORMULAE
521
but is fairly close. Now if we could lay off all three sets at
precisely equal angles the z-scale would become easily com-
mensurable, and the scale-modulus of the z-scale could be used
like the other scale-moduli.
There is much to recommend such an arrangement of trl-
linear co-ordinates. All intersections would be distinct, hence
greater accuracy would be achieved in the use of the chart.
The useful portions of the chart would be more compactly
positioned, hence space would be conserved and greater detail
available. The form would be unusual and more attractive,
an important feature, since, as we shall presently see, these
charts are more pictorial and popular than business-like. Yet
in spite of these advantages, the equilateral and equi-angular
form is seldom or never used, probably because the average
chart-maker has become so accustomed to rectangular co-
ordinates.
The additive chart which we have described can be turned
into a factorial one by logarithmic projection of scales. It
can then be used to calculate the formula Y = since log
LABOR TIMS (@ BO^ per hour)
CURVES FOR FORMULAE
523
Y =log k-\-a log X+c log Z. The chart is prepared precisely
as is the additive chart, and is much more generally useful.
Z » XY
Fig. 406.
An exponential chart can be obtained by the combination of
logarithmic and arithmetical scales, for the formula Y = aFZ
and the like. Other possible combinations will occur to the
student. The chief use of the chart, however, has so far been
in its additive and factorial forms, having arithmetical and
logarithmic scale projections only. With these two the student
should be thoroughly familiar.
We come now to another chart which has straight-line
z-curves or diagonals. It is easily distinguishable from the fore-
going from the fact that the z-curves are not parallel to each
other, but radiate from a common intersection point. The
parallel line chart was simply a multitude of curves for the
linear equation, Y = aX-\-C, in which many values of C were
taken and C itself treated as a variable. The radiating straight-
CHARTS AND GRAPHS
524
line chart is simply a multitude of curves for the same linear
equation, save that many values of A are taken, and A is
Plotting Fcrsiula
Fig. 407. Chart for Determining Scales of Curve-charts.
At the bottom of the chart find the value of the highest point in the curve. At
the left-hand side find the height which you wish to give it on the chart. At the
nearest intersection of the ordinate with a diagonal at this height read (in the
scale of diagonals above and to the right) the side of the rule, N, and the plotting
value of each unit on this scale (U).
treated as a variable. For convenience we will still call this
third variable Z, and the equation therefore which this chart
solves is of the form Y =ZX, when the common point through
which all diagonals pass is the origin of the x- and y-axes.
When the common point is located elsewhere on the y-axis,
as at the co-ordinates ;f=0, y =r, the chart solves the equation
Y=2X+c. The chart is primarily factorial, though it has
CURVES FOR FORMULAE
525
arithmetically projected a:- and y-scales. The s-scale is a scale
of angles, a circular function of the a;- and y-scales.
A?
Fig. 408. •
We shall not go into the principles of this chart in detail,
it can be easily made, the x- and y-scales being arithmetical
and the z-diagonals and z-scale being best put in by inspection.
The chart is not of much value (save for one particular pur-
pose), it is difficult to use accurately when the values must be
interpolated between co-ordinates and diagonals and loses
detail as the diagonals converge. For the processes of multi-
plication and division which are the main purpose of this arith-
metical factorial chart, the logarithmic factorial chart is in
every way, save one, the more satisfactory.
There are a great many other calculating charts upon co-
ordinates, which have not the straight line diagonals or z-curves,
but in which the third variable is shown literally by a series
of curved lines, each bearing a particular value of Z. These
are, of course, only multiple parabolic, hyperbolic, or other
CURVES FOR FORMULAE
527
curves (even including circles and ellipses), and solve more
complicated equations than the forms already discussed.
Fig. 411.
They are largely of academic interest, however, forming in-
teresting exercises for the student and somewhat highly pic-
torial displays of the behavior of phenomena to which their
formulae apply. For practical purposes they are of little value,
since they take up much time and effort in the making and
are neither so accurately nor so easily used as the calculating
charts to which we shall later come.
We come lastly to the multiple calculating chart, a com-
bination of two or more simple charts of the types described.
In the instances considered, the charts have shown only three
variables, and therefore been suitable only to equations with
two variables beside the root of the equation (in Itself a vari-
able but dependent upon the other two variables). More
complicated formulae, with three, four, five or more, inde-
pendent variables, can also be shown by these calculating
charts, by the simple trick of joining together a number of
individual charts. Thus the first chart can show two*inde-
528
CHARTS AND GRAPHS
AUGNMENT CHART FOR SOWT/OR OF CUADRATW ANO CUBIC BCUAT/C/VS.
From '•Graphical and Mechanical Computation'* by Joseph Lipka, published by John Wiley &• Sons*
by permission.
Fig. 412. A More Complicated Chart for Solving Quadratic and Cubic
Equations.
The presence of this chart in this chapter was discovered too late to shift it to its
proper place m the chapter on Composite and Zigzag Nomographs, about page 576 ,
CURVES FOR FORMULAE
5^9
pendent variables on its x and z-scales, and their resultant
upon its y-scale. The y-scale of this chart can be used as the
x-scale of a second and adjoining chart, a third independent
variable appearing on the z-scale of this second chart and the
new resultant on its y-scale.
By continuing to join new charts to the old ones, the number
of variables in the 'equation can be increased indefinitely.
Where all the independent variables are factorial, we can use
the logarithmic projection throughout with parallel straight-
line z-curves on every chart, but where some of them are
additive, it is not possible to use the logarithmic projection.
It is for such cases that the arithmetically projected factorial
charts last described come in handy, as by their use the addition
processes can be performed upon additive charts and joined
530
CHARTS AND GRAPHS
to factorial charts through the common arithmetically pro-
jected scales. Often, the various charts are not set side by
frftdjECTEd AREA OP CROSSHEAO figAftlNC SURFACE IN SQUARE INCHES
From E. A. Andrews, in **Machinery” and HaskeWs **Ii<yw to Make and Use Graphic Charts.”
Fig. 414. A Composite Chart With Many Scales.
Showing Loads on important engine frame members.
side but are superimposed, the reader being asked to follow
a sort of mystic maze along these ambiguous co-ordinates till
he arrives safely upon the “home”-line scale and meets the
answer to his problem awaiting him there.
All of these charts are more sensational then satisfactory.
Needless to say, their preparation consumes much time. A
large amount of excellent zeal is sometimes displayed by in-
experienced chartists in the formation of beautifully-drawn
and elaborate chart-forms, suitable for equations with many
variables. It is always disappointing to observe the beautiful
work and the great energy which has gone into the prepara-
total load on CBOSSHEAO guide !N pounds
532
CHARTS AND GRAPHS
curve equational chart forms are uselessly wasteful of time and
energy. Everything that can be accomplished by these elab-
orate and beautiful charts can be accomplished much more
simply, accurately and easily by the use of the charts which
will be described in the following chapter, in which the intri-
cate network of co-ordinates® and curves alike is entirely omit-
ted and the scales alone are presented upon paper utterly de-
tached from their fields and curves.
^ It is indeed true that the co-ordinates need not be used on the curve if rectangular
movable axes (similar to isopleths) on separate transparent sheets be used to project
'■he co-ordinates of the point to the scales where they may be read# .
Chapter XLVI
PARALLEL NOMOGRAPHS
In the calculating charts just discussed, we noted that
the values which solved a mathematical equation lay along
a curve and that the chart was more easily constructed and
accurately used when these curves formed straight lines. From
this last condition, it is but a step to conclude that the straight-
line curves themselves could be omitted and a movable
straight-edge (ruler-edge or tightly drawn piece of thread)
could be used in their stead, the reader of the chart being re-
quired to adjust the straight-edge afresh for each reading.
The only objection to this step is that while the lines are
straight lines, their angles are arbitrarily set by the problem
and the straight-edge must be adjusted at a certain angle or
slope before it can be used. But in the charts which we will
now consider, this obstruction is removed, the charts being so
designed that interpolation by means of the straight-edge is
possible in any and every position of the edge, the equation
being satisfied always. The straight-line transversal is no
longer called a curve, but is now known as an “isopleth,” the
points through which it passes being always of equal value,
that is, forming an equation. The scales are now called “axes”
in a wholly isolated sense. The chart itself is called a “nomo-
graph,” “nomogram,” or “alignment ch art,” the latter name
being obviously derived from the fact that the proper corre-
sponding values are always in perfect alignment.
In the nomograph, the network of co-ordinates and the
plotted curves themselves being omitted, there are three
scales alone retained. These scales are the scales for the two
axes of the curve and the added scale of the diagonal curves
themselves.^ But the scales are so carefully arranged, both as
to their projection and as to their position, that the intersection
of a straight line or isopleth across two scales always gives the
proper corresponding value upon the third scale. The arrange-
533
5J4
CHARTS AND GRAPHS
merit of these three scales is either parallel or zigzag, so that
nomographs can be divided into two classes, the parallel and
the zigzag nomographs. Manyi other forms are possible, but
are largely of academic interest. The two principal forms are
the simplest and most satisfactory for all work.
The parallel nomographs are based upon the geometrical
theorem of similar triangles. If we take the simplest case, in
which the three parallel axes or scales, which may be called
the X, y, and 2 scales, or axes, are so arranged that the two
outer ones (let us say the x and z axes) are equidistant from
the middle or y-axis, then, we will see that whenever two lines
(isopleths) cross these three axes, the distance laid off on the
middle axis will be the average (arithmetic mean) of the two
distances laid off on the outer axes, between these cross lines.
To make this quite clear let us set three rulers up on end
against the wall at equal distances along the wall. With the
rulers resting on the floor, their zero-points or lower ends will
be in a straight line, the line of the floor itself. Note that
the floor here forms an isopleth, the average of the two outer
zeros being shown by the middle zero. Now if we hold a
piece of string tightly stretched across these rulers, we will
^Cf. Lipka, Joseph, Graphical and Mechanical Computation, John Wiley & Sons.
Peddle, John B., Construction of Graphical Charts, McGraw-Hill Book Co. 'dnd
Running, Theodore R., Empirical Formula^, John Wiley Ik Sons.
PARALLEL mMOGRJPlI^
-
see that the value on the middle ruler always equals one half
the sum of the values on the outer rulers. If the string passes
4U + 2)
Fig. 417.
across the first or ^-rule at the'point of six inches, and across
the third or s-ruler at the point of ten inches from the floor,
it will obviously cross the middle or y-ruler at the point of 8
inches, one half of the sum of six and ten. The general form
of the equation is
y = or 2 y ~x-rz.
To adapt this device to the processes of addition and sub-
traction is a simple process. Let us merely substitute a ruler
calibrated to half-inches for the inch-rule in the middle. In
other words, let us substitute for the y-scale a scale with values
of Y such that each value of Y is just twice as large a number
as its actual y-distance, that is, T = 2 y. Now the readings Y
on the y-scale will be, not the average, but the sum of the
readings on the jr and z scales. The formula for the chart be-
comes 2y—x+Zy or Y — X+Z. And for all positions of the
cross-line or isopleth, the intersected points Yy X, and Z, will
have the relation F = X+Z. Subtraction may obviously be
53 ^
CHARTS AND GRAPHS
performed on such a chart by adjusting the straight-edge or
isopleth through any given values on the X and Y or on the
X ? *
S - X "i* X.
Fig. 418.
Z and Y scales, the difference being shown on the remaining
(or other outer) scale. For if Y = X+Z then it is clear that
X = y — Z and Z — Y —X. The middle or y-axis always carries
the minuend in this arrangement, just as it always carried
the sum when the same arrangement is used for addition.
Before going further let us again carefully take stock of the
algebraic symbols which we shall use in this chapter.^ The
chart, as we have seen, uses one or more sets of three axes,
which we shall call the {x)y (y), and {%) axes. Along each
of these axes we measure distances, y, and %y in terms of
various units of length, or scale moduli,^ niyj and all of
which are readily convertible into a common unit of length,
^ See also previous chapter.
® Throughout the formulae for nomographs in this book, the scale-moduli War, my,
and mzi have been used to indicate the plotting instructions for the variable X, F,
and Z. These formulae, therefore, differ slightly from those of Professor Lipka, in
whose book the scale-moduli are used to indicate plotting instructions for the func-
tions of the variables, /(.v), /(y), and /(x).
PARALLEL NOMOGRAPHS
537
or chart-modulus, m. The values which are entered or cali-
brated at these distances are X, Y, and Z. These last, X,
X
Y
Y « X2
Fig. 419.
Y, and ZT are’the symbols of the variables in the equation
plotted, they are the scale-figures which appear on the chart,
which afford, by their readings along the isopleth, the solutions
to the equation.
To adapt this device of three parallel scales to the proc-
esses of multiplication and division we need merely change
the calibrations on the three scales to a logarithmic projection.
As always, the actual distances, x, y, and z, on the three axes,
still have the relation 2 y =x-fz. But we plot on the x-scale
the values of log X, on the z-scale the values of log Z and on
the y-scale the values of | log Y, using the plotting equa-
tions, x — m logX, z = 7?ilog Z, and 2y =ra logT or y = y log Y.
Since 2 y =x+z, log Y =log X -flog Z and Y = XZ. The read-
ings of Y are the product of the readings on X and Z, and
multiplication is accomplished by adjusting the straight-edge or
isopleth through given points on X and Z and reading their
538
CHARTS AND GRAPHS
product on Y. Division, like subtraction, is accomplished by
adjusting the isopleth through points in the middle and one
outer scale, the answer being read on the other outer scale. For
if Y = XZ, then and
The middle or y-axis al-
ways carries the dividend and the product in this arrangement.
As it will be seen that the distance on the middle scale is
always the average of distances on the outer scales, we must
expect normally the resultant, that is, the sum or minuend
(arithmetically), or the product or dividend (logarithmically)
to appear on the central axis only. But a rearrangement of
scales can be made when it is desired to place this variable
on an outer axis. For this we must use complementary num-
bers in addition and subtraction, and reciprocals in multiplica-
tion and division. That is, we must substitute 0 -X, for X,
in addition, and — for X or 0
X
-log X for log X in multiplica-
tion. In this arrangement of the additive nomograph (for
additions and subtractions) we make x=m{ ~X) or -mX.
Then since 2y =;c+y, we have the equation Y = ^=—-\-XL =
m m m
-X-bZ = Z-Z. And since Y = Z -X, then Z = r+X and
X — Z— Y. In short by upsetting the scale on the first axis
we have exchanged the meanings of the scales on the second
and third axes and the third axis now is the resultant (sum or
minuend). The rearrangement of the factorial nomograph
(for multiplying and dividing) is similar. Here we make
x — miO -log A), with the result that -log X =— = ^ — — =
m mm
log Y -log Z and log Z = log Z -log Y and Z=— or Z = XY.
The same effect is noticeable as before, the upsetting of one
outer scale shifting the meaning of the other two scales, the
other outer scale becoming the resultant (product or dividend).
Writers on nomographs are accustomed to attach import-
ance to the position (vertically) of the scales along the axes,
a detail to which the cases of reversed or upset scales which
we have just considered, naturally leads us. It has been as-
sumed in this discussion that you have kept in mind the idea
of three rulers stood up on the floor against the wall, for this
makes clear that the three axes must have a common base
PARALLEL NOMOGRAPHS
539
line, or isopleth passing through their “zero-distances” (regard-
less of the calibrations which may be assigned to these dis-
tances). Thus when the scales are reversed as in w( -bg X) =
or log X , it is obvious that we are changing the di-
m
rection of measurement and counting downward into or below
the floor level and an isopleth across the three rulers would
have to pass up through the floor. Now it is not at all necessary
that the base line (i.e. floor line) be at right angles to the axes
(or rulers); our line passing through common ^^zero-distance'^*
points or ‘^origins’’ of the axes can be a very steep diagonal.
And when one of the scales has been reversed, it is distinctly
better to use a diagonal base-line so that the isopleths used in
solving the problems by the formula of the chart, shall be as
much as possible at right angles to the axes, to facilitate
accurate readings.
540
CHARTS AND GRAPHS
Frequently, in fact, more often than not, the values in which
you are interested do not begin at zero, but begin at some
distance up the scale, that is to say, the useful or desired range
of the variables does not come down all the way to the base-
line or origin of the axes. In that case again, it is well to use a
U+ll
12
•12
13 - 4-13
38 -
-38
37 -
-37
36 -
-36
36 -
-36
34 -
"34
83 -
• S 5
32 -
-32
31 -
■31
30 -
-30
29 -
-29
28 -
-28
27 -
-27
28 -
-26
26 -
-26
24 -
-24
23 -
•23
22 -
-22
X
y
e6y58
67- -67
63 -
62 --62
61" -61
60 -
•60
t » Z - 2X
Fig. 422.
diagonal base-line, in order that you may omit the lower parts
of the scale entirely, together with the base-line itself, on your
PARALLEL NOMOGRAPHS
541
finished chart. Another way to achieve the same result is to
alter the calibrations alone, so that x = m{X-]r a), y=^m
(Y-tb) and z = ot(Z+c), (in which a, b, and c are constants
which in themselves satisfy the formula). In general, the
values of these constants should be such that they are equiv-
alent to the lower limits of the desired ranges of the variables.
The real object of the diagonal base-line or diagonal zero
isopleth, is to make all isopleths which will be used on the chart
as perpendicular to the scales as possible. The nearer to a
right-angle the intersection of isopleth and axis becomes, the
more sharply the two lines cut each other and the more easily
will accurate readings be made.
This brings us to the important element of the range of
the variables. For it is not necessary, nor even possible, to
picture all the possible values of a variable upon a chart. In
actual problems the independent variables will usually be found
to fluctuate between certain limits. It is thus unnecessary to
use a scale so great that it shows values in excess of the maxi-
mum limits, or to include on the scale the values below the
minimum limits. Space is conserved and detail gained by
making the range of the scale conform to the range of the
useful values of the variable. And when the ranges of each of
the two independent variables have been set, it is easy to find
the range of the resultant or dependent variable (the root of
the equation). In the previous chapter we have indicated a
method of noting these limits, thus
Y = X
10
0
+ Z
5
0
, or Y
15
0
10
0
+ Z
5
0
A variety of methods are at hand for confining the chart to
these ranges. We may place the lower limits at the zero dis-
tances or origins of each axis. Or we may place the maxima
upon the level (or base-line). Or best of all, we may place the
mid-points along each range upon a level isopleth. The
advantage of the last method is that all the possible isopleths
will then cross the axes at angles nearer to a right angle than
by any other arrangement. Having approximately positioned
our scales with this object in view we do not actually need
to calculate the values of the mid-points (fractional as these
may be), for plotting; we need only calculate the values for
any round numbers and precisely position the scales about
them.
54 ^
CHARTS AND GRAPHS
An important point in the making of the chart is its total
size and proportions. Both its height and width should be
great enough to serve whatever purposes of convenience in
use, legibility and detail of readings, visibility at certain
distances, or success in reproduction and reduction, will natur-
ally obtain in chart-making, but the width should always be
at least as great and if possible half as great again as the height.
If the chart is too narrow many of its useful isopleths will cross
the axes at such small angles that correct readings are difficult.
If the chart is too wide, the isopleths will all cross at very good
angles but the scales will be so closely compressed as to make
detailed readings hard. The best form in general is one in
which the most steeply sloping isopleths cannot cross the axes
at smaller angles than from 45 degrees to 60 degrees. The
height should be approximately two-thirds the width.
The final consideration is the choice of axis for the de-
pendent variable. By the dependent variable is meant the
variable whose values are sought from given values of the
Fig. 423. The Inverted X-Scule.
PARALLEL NOMOGRAPHS
543
other variables. Of course it often happens that the same
equation is often used backwards, and that at times one vari-
able is sought from given values of the other two and at times
another is sought. But usually there is one variable which is
most likely to be the unknown and this should be treated as
the dependent variable. The best axis for the dependent
variable is always, ceteris equibuSy the (3^) axis. For then all
needed isopleths will lie within the limits of the two outer
scales and the three scales can be of roughl}?- uniform height.
Fig. 424, The Use of an Outer Scale for the Unknown
Variable is Not Good.
544
CHARTS AND GRAPHS
Were the known variable placed upon an outer axis, it is
clear that it would have to be extending above and below
the levels of the other axes unil it included the most ex-
tremely sloping isopleths which could be drawn through the
central and other outer scales. The result would be a chart
of very irregular appearance, wasteful in space and involving
less accurate readings because of smaller angles of intersection
between isopleth and axis. The danger of errors in placing
the isopleth would be four times as great, since the errors in
positioning the known values may be doubled upon the un-
known scale, whereas they are halved when the unknown scale
is on the central axis.
It has been the purpose of the foregoing discussion to be
suggestive rather than definitive, of the general principles of
the parallel nomograph. It remains to examine this chart
analytically. This will lead us at once to a generalized form
of the parallel nomograph, with important modifications which
make it far more flexible, in use. The chart has so far been
considered only with equidistant axes.
The geometrical proposition of similar triangles can equally
be applied to axes which are not equidistant. When the inter-
val between axes (x) and (y) is equal to that between (y) and
(s), then the formula for actual distances is, as we have seen.
y =-2 +*2 ^ y==x+z.
And if we denote the total distance
Fig. 423.
PARALLEL NOMOGRAPHS
545
between the {x) and {%) axes, either measured perpendicularly
to the axes or along the base-line or along any isopleth, as
'^p+qP taking ‘^p'^ as the part between the (x) and (y) axes
and as the part between the (y) and (z) axes, we may write
the formula for the distances or axes cut between isopleths as
Ip + q)
(p)
iP + q)
PLshPl
P + q
or (p + q)y = gx + pz.
This formula is applicable to any parallel nomograph, no
matter at what distances from each other the axes may be
placed. And the significance of '‘p'' and ^^q*' are very easily
seen. They are the coefficients of the x and z variables in the
additive formula 2 Y = A" + Z, and the corresponding expo-
nents in the factorial formula = A"Z, becoming coefficients
in the corresponding equation 2 log Y — log X + log Z. In short
the complete formulae are, for additive charts (p + q) y = qx +
pz, and for factorial charts, y + z^. Obviously when
p and q are equal they may be written as 1 so that w^e have
2y ^ \ X + I z and y^ = z^. So that the formula at once
explains the half-size scales taken for the middle axes. Also
when w^e reversed or upset one scale, w^e were in the additive
formula inserting a -- 1 coefficient, making the value of ^ — 1.
We were then obliged to shift the other outer axis into
position midway between the first and second axes (a process
which we spoke of as exchanging meanings of scales) so that
p became + 2 and {p -f- q) became + 1, so that the formula^
became ly = — 1 x -{-2z. Thus the general formula {p + q) y
= gx + pz covers all cases of the parallel nomographs,
We are now ready to lay down the rules for the construc-
tion of the parallel nomograph. In the first place, we have an
^ Or, caliing the y-axis s, because it is now the third, and the s-axis y, because it is
now the second, we have 4- Is == — l.t -j- 2 y, which agrees exactly with the formula for
reversed scales. So doing, we maintain the symbols (x), (y) and (s) for the axes strictly
in the order in which these axes appear on the chart.
It is obviously better to permit the symbols (x), (y), and (z) to adhere to the axes
wherever they appear, regardless of their older upon the page, as the general formula
then applies consistently and without confusion. In the text from this point on, this
has been done, and the (xj-axis need no longer he the first, the fy)-axis the second,
nor the (zj-axis the third; but the algebraic signs of p and q will signify changes in
position, and the algebraic signs of the scale-moduli will be significant of the direction
of plotting.
CHARTS AND GRAPHS
546
TT TT
equation of the general type Y=AX -^+CZ + K, which®
we wish to present upon a chart or diagram which has the
relations of gx pz = {p + g)y. We give this diagram any
height, Tm, we wish and approximately half as much more
width. If, as is most convenient, we let the chart-modulus,
m, equal 1 inch, then T is the total height of the chart, or
length of each scale, in inches. Now along these scales we
propose to plot the values of the independent variables, X
and Z, from their lowest, L, to their highest, H, useful values.
Call the difference between these extremes the range, R, of
the variable, then
R,=H,-U
We can easily plot the values X and Y through these
ranges in these given lengths by the method of triangulation
if, as is generally the case, the intervals are not even fractions
of the inch.® Then draw an isopleth through any convenient
values of X and Z on the (x) and (z) scales and we know that
the corresponding value of Y lies somewhere along this iso-
pleth. Substitute these values of X and Z in the equation and
learn the corresponding value of Y. Select a second conveni-
ent value for X and substituting it and the T-value for X and
Y in the equation, solve and get a second corresponding value
of Z. Draw a second isopleth through the second values for X
and Z on the (x) and (z) scales and since the value of Y has
remained unchanged, we know that the (y) axis passes through
the intersection of the two trial isopleths, and is parallel to the
other axes. Now solve a few more equations containing con-
venient values of X and Y and draw their isopleths and you
will rapidly calibrate the (y) axis with its Y values. After a
few points have been plotted the rest of the I'^scale can be
put in by a ruler, and the method of triangulation. If these
directions are carried out the entire chart will be finished in a
short time.
The student will look however, for an analytical method
which will define mathematically the various scales and their
® Within the short vertical parallel lines in the equation are inserted the high, //,
and low, Z, values of the variables which will be required. These maxima and minima
of the ranges are merely memoranda which do not affect the equation in the least,
and can be omitted from the equation and noted elsewhere, if they render the equation
confusing.
For the precise adjustment of scales to given sizes by the method of tri angula-
tion, see Chapter XVII, page 185.
fiiRHlEv HOWOSHiPK;
I OWAf ( cut
5-^7
» • AX * C2 ♦ K
5w - Jy « S*
4S9«SNU£Mr »0 «viflot.«:
TO » «Bf »€ «»0£ KT ««RI«StC»: Lit X •» «0
1 •» •a
TO OIRfKOCWT T*BI»gtf; V «* BU
imirs or us(fuu v»ni»riOMs:
CilcutAT lOMs roK Y->«C4t.i:
ditEN X - 0 4H9 Z » SO ; V » 1312)0 * {3)2)20 - 30
‘i ^ iO I • 14 . X • (2)3)50 . i6f»)U • U
\ • 49 2— IS ; X • (2/3)45 - (6)3)15 • 6
T - J5 2 • 13 : X • (2)3)26 - (6)3)13 • -3
Ijr IJ# Z • IS ; X • (2)3)115 - (6)9)19 •
^ » IT
Fig* 426. Construction of the Parallel Nomograph— I.
Finding the unknown scale by trial isopleths.
54.8
CHARTS AND GRAPHS
positions. For this we must know the scale-moduli, w., and
as distinct from the chart-modulus, m. The scale-modulus
is the interval or unit distance along the scale between the
unit values of the variables, X or Z, and is obviously written
=
R.
■ m
or letting w = 1 inch)
= inches
Rx
■
L
R.
inches
Now in plotting it is always a convenience to adopt units
of length such that they can be laid ojfF from an engineer’s
hexagonal rule and do not have to be specially projected by
triangulation. The engineer’s rule divides the inch into 1, 2,
3,4, S, or 6 parts and decimals multiple or submultiple thereof,
such as 10, 20, 400, 3000, etc. If we describe the scale moduli
by the numbers of them which go to make up the chart modu-
lus, My and call this number S, then
Sx'fUx = ni S^nt. = m
and
m
m. =
S,
or, letting m —1, (that is, one inch)
1 1
m,= —
Thus as we have previously seen, the reciprocal of the scale
modulus, when m = 1 inch, is the number of intervals per inch
and serves as an index of the proper side of the engineer’s rule
to use in plotting. Combining the two equations for we
can eliminate it and keep measurements in terms of its recipro-
cal, S, thus
m
m
or
El
T
If S in these equations becomes either 1, 2, 3, 4, 5, or 6,
or any multiple of any power of ten, we can of course
PARALLEL NOMOGRAPHS
549
plot the {x) and (y) scales directly from the engineer's rule.
If it does not do so at once (and it usually doesn’t) always
make it do so by altering the values of R and T slightly; that
is, change the length of the scale, 7", or increase its range, i?,
or do both. Slight alterations of this kind do not appreciably
affect the size or usefulness of the chart. We shall write these
altered values in small letters, thus
So we see that by selecting values for r and t which are
close to the original values R and T, but which make S pre-
cisely indicate a side of the engineer’s rule, we make the chart
even simpler to draw.
Now we have seen that the chart has the linear relations,
qx+pz = {p+q)y. If the chart is to express the equation,
^AX ACZ-{-K^ and we decide (since it is easiest) to correct
for the added constant K along the (y) scale, then we write
AX+CZ^Y
and the chart must have the values
qx = 771 (. 7 . Y ) pz = 771 (CZ) (p +q) y = 771 {Y -K)
1
p+q
771 {Y ^K)
In this the distances, a;, y, and t;, are taken fiom origins
which will be real if and Ly are zero, but imaginary
if La:, and Ly are other than zero and the zero or base-line
is not shown on the chart. Also we know that by definition
a ; = niy^X z = 17 x 2 . y = my {Y -K)
Hence
A C
fUx = 771 = 771
q “ p
But from above
7n,y
1
P+q
7Yl
77fl
Hence
1 A
1 C
550
CHARTS AND GRAPHS
PARALLEL NOMOGRAPH:
SYMBOLS, LIMITS, AND DEPENDENCE
Hx
AX + CZ
3 45 5 20
_ X + - Z
2-5 2 12
SIZE OF CHART (HEIGHT): LET T = 4 inches
RANGE OF VARIATION:
- 74 - - 50 = TTj - Ij. == S
RULERS FOR PLOTTING:
^ '’'x /
50 / 4 = 12.5
50 / 2.5 = 20
50 / 5 = 10
INTER-AXIAL DISTANCES:
(3/2) 10 = 15
Fig. 427. Construction of the Parallel Nomograph— IL
Finding rulers with which to plot the known variables and a formula with which
to position the unknown axis. (The ruler values adopted are underlined in the
worksheet above.) The unknown variable is still plotted by trial isopleths.
^3 — H / h
8/4^2
^ = (5/2) 2 « 5
EQUATION:
JX-^CZ-hK
2u'= 3v 4 - 5w
® Y-K
PARALLEL NOMOGRAPHS
SSI
Here we have a convenient formula for the distances, p
and <7, between the three axes, (.1:), (y), and {%). Having found
from the range and total length of the scales the convenient
sides of the rule to use in plotting them (S— — ) we merely
multiply these by the coefficients, A and C, of the variables in
the equation to get the horizontal distances between axes.
You will notice that the chart-modulus, m, has cancelled out
of the equations, so that p and q can be measured in any units
which will make their sum be the desired width of the chart.
These devices have obviated the first two trial solutions of the
formula for the purpose of locating the (y) axis.
Lastly we come to the formula for the plotting of the (y)
scale itself.
m
Just as we wrote m^=*^and
m
so we can
write My—— and our object will be to find Sy, such that it
Oy
t(»o can be plotted directly from an engineer’s rule. Above,
we see that
p+q
niy hence^
Sy =p-hq ^ ASx'i‘CSg
A and C, of course, are fixed, and you will often find it im-
possible to so adjust Sx and that while they indicate sides
of the engineer’s rule, Sy does the same. It is generally neces-
sary, to go back to the original elements of and Ss, namely
f T tr
— and — , and alter one or both of them until the desired
tx tP
result is achieved. A convenient plan is to set down columns
for each of the elements, Sy, py S., r^, and 4, so
that you can try a number of different values of T and J?,
for each variable before you give up hope.
When Sy cannot be made to conform to a side of the rule,
it is still always easy to project specially by the method of
^ Since obviously
552 CHARTS AND GRAPHS
P/^RALCEL NOMOGRAPH: eouation: 2 u * 30 * 5 m
PUOTTIMa I tSTHUCT lOMft:
H
8x
9
H
n
H
iSx
q*p
esg
li
V^X*50
‘■1
0-1
Rg-a
6
50
•5
^4
1
^3
$
8
5
50
10
15
30
0
2
8
4
2.5
50
20
30
a;* 5
716
3
9
3
2.5
50
20
30
dO
10
4
8
2
8.5
51
8
9
11.5
2.5
1
8
$
10
50
6
7.5
9
1.5
9
15
10
50
5
7.6
10
2.5
1
9
i
Figr- 428. Corvitruction of the Parallel Nomograph — III.
Finding ruler-values with which to plot the unknown variable. (Any of the sets
of underlined rulers for the three variables can be used, according to the size,
Ta? and Tz, which is desired for the chart.) Worksheet only is shown here,
triangulation. There is no more need to make trial isopleths
for specially computed values of the variables, and we would
only make two or three of these when the chart is completed
to check it up for accuracy. The whole problem of charting
the equation Y =AX p +CZ is reduced to the follow-
Ltx -Ltz
ing simple steps:
YCZ^\ = Y -K
hx L,z
I\ -IRJLIJiL }JOMOGR. n^HS
553
Sx=^ when Tx approximates = Hx —Lx.
T
Ss = -f when approximates R^ = Hx -L^.
p = cs,, q - AS,, and Sy = CS,+AS,.
We know T, the height which we wish to give the chart
and which t, and approximate, and we know the width
3T
which we wish to give the chart, approximately — , which will
be divided by the axes in the ratio of p and q. It only remains
for us to select mid-points in the ranges of X and Z and place
them, with the corresponding value of Y, upon a common
horizontal isopleth and plot the scales about them.
When the equation is factorial instead of additive, it has
the form
or
log Y^A log X
X
^ log X
-\-C log Z
I-hog.
^log :
+log K
and the same treatment may be followed precisely. It is
more convenient, however, to plot directly from a log rule,
if one is handy, than from an engineer’s rule and a table of
logs. We therefore drop the engineer’s rule and use the cali-
brations on a slide-rule, if one is available. The simpler slide-
rules have two scales, one a single and the other a double deck,
in the length of 25 centimeters. Better slide-rules have also
a three-deck scale. Taking the modulus, m, of the chart as
25 centimeters instead of 1 inch (it does not make any differ-
ence in the planning equations just listed since m has been
cancelled out of them) we now measure T, f, and f, in units
of 25 centimeters, roughly 10 inches, and take S == 1 to indi-
cate the single, S == 2 the double, and S = 3 the triple deck
scales. In short, when S,, S^, or Sy can be made equal to
1, 2, or 3, we can plot scales directly from the slide-rule.
Complicated formulae cannot often be made to yield
direct ruler-copying values of all three, S,, S-, and Sy, at the
same time, either for the additive or the factorial charts.
This difficulty is most frequently encountered in the factorial
charts because of the more limited number of different rulers.
When this is the case, the method of parallel triangulation
can, of course, be used for all other values of S,, S^, and Sy.
But most convenient of all is a set of radiating triangulation
5S4
CHARTS AND GRAPHS
sheets, such as are included in Professor Lipka’s book,® which
can be folded at any value of m and will give all possible pro-
jections of the arithmetic or logarithmic scales. When such
devices are used the chart-maker has no occasion to seek
certain values of S, but can work with any scale-moduli what-
ever, so that his equations become (when m, the chart-modulus,
is 1 inch):^
T T
w* = — w.=
A,
? =
m.
i> =
R,
C
= -
1
VI.,
p+q
and he can plot directly from his sheet of scales, folded at the
proper scale-modulus.
The general equations which have been given, namely.
Y = AX'
and
L.
\H.
L.
rm
1+Cz|g+A-
'W'
14
are usually found in simplified form, K being 0 in the
additive (first) form, or 1 in the factorial (second) form.
When a coefficient (in the additive) or exponent (in the fac-
torial) is attached to the dependent variable, Y, it can be
transferred to the other variables so as to clear Y, by division
or involution. When the signs of either X or Z are negative,
the sign must be treated as part of the coefficient and so trans-
ferred to the scale-modulus or ruler-index (S), to indicate
that all values are plotted downward instead of upward.
• Variations of this parallel nomograph will occur to the
student, such as charts for the equation
Y = X-^7^K + D,
which must be turned into
log {Y -D) -log K =A log X+C log Z
* To the chart-maker. Professor Lipka’s book Graphical and Mechanical Com-
futaiim is well worth its cost, if for no other reason than for the useful scales it
contains in a pocket in the rear cover of the book. These scales carry radiating lines
Irom a common center to all parts of a ten-inch uniform and a ten-inch logarithmic
(single-deck) scale. By folding these sheets appropriately, these scales can be obtained
from the radiating lines at any desired smaller scale. They amount to complete
outfits for the triangulation method of scale adjustment,
® As is the case with Lipka’s chart-formulae.
PARALLEL NOMOGRAPHS
555
NOMOQRAf^H: COUATidii:
3T.JS n m
13* f 11 g^
Hx
AX 4
cz
■ Y - «
25
50
3 lot I 4
.25
a log Z m
*05
•
tog T • log 5*17
CMAAT IM6
iNarmiet iONs:
T
* ¥ tnoAe«
Rjj • lot SB - lo4 ,U5 « lot 100 Rg *' • log lOOiS
■ 1 » 3
«X “ 3 Incftf# «2 “ “ </3 - 2 tndA*«
^ - A/*, • 2/a • .66? p - C/»2 " 5/2 - I, 5
•X * 3 /(P*<l) " inchBH
¥
I
100,000,000“
noo, 000,000
10,000,000-
no.ooo.ooo
l,ooo,orci-;
-1,000,000
100 , 000-2
1
=“100,000
-1
10,000-
-10,000
1,000 -z
yl,000
100-
-100
—10
1-
— 1
c
H
.01 -z
^.01
.001
f.OOl
.000,1-:
r-.000,l
Fig. 420. Construction of the Factorial Parallel Nomograph.
Finding slide-ruler plotting scales for all variables and locating the unknown
variable axis, all by formula. (Only the adopted values are shown in the work-
sheet, but a columnar form similar to that used in the last figure, is useful to corn-
par/ different values before selection.)
SIZE OF TYPE IK POINTS
556 CHARTS AND GRAPHS
Fig* 430. Chart for Determining Size of Type.
Showing the space required to^set up manuscript in specified sizes and styles of
type. The latter have been indicated on the chart on a properly graduated
dummy axis, but without numerical calibrations, of characters per inch, as these
would be useless.
PARALLEL NOMOGRAPHS
SSI
and involve therefore a special logarithmic projection with
shifted zero. The exponential equation
Y = A^C^Ki
can be turned into
log Y -log K = (log A) X+(log C) Z
and involves a mixture of log and arithmetical scales, the
scales of (a;) and (z) being arithmetical. So does the equation
Y = A^Z^K.
A log-log projection is called for by the equation
Y = AX^^
which turns first into
log r = (log ^)-l- CZ (log X)
and then into
log (log Y -log A) =log C-flog Z-p log-log X.
The projection of powers, roots, and reciprocals all find occa-
sional use. Indeed any function whatever of two variables
■ -I
7.6 -
e.6
* H
1.6H
iK-60 e 0- 1.
v-io e u-io
iii-to ® 0 - 1 .
]t.60 <3 D- 1 .
K-60 © 0. 1
H-40 ® 0- 1
N-30 « U- 1.
2K*50 ® U- 1.
H-2C ©0-1
2N-30 ® U- 1 .
i«.60 ® 0- OO
H-10 ® 0- 1.
® D- 0.1
5 N '60 ® 0 - 0.1
fi W-SO ® U- 0.1
^ M-40 ® 0- O.X
It
3 K-30 ® 0- O.l
^ ih-50 9 U- O.l
H.20 ® U- O.l
2K-30 ® U- 0.1
i «-60 ® 0 - 0.01
K-10 ® U- O.l
iS-40 ® 0- 0.01
M-60 ® 0- 0.01
|f-50 ® U- 0.01
K-40 e 0. 0.01
H-50 ® 0- 0.01
li
li
a
**
St
4
4
6
K +
g_>- •
Fig. 431. Chart for Determining Scales of Curve-charts.
On the left-hand scale find the greatest value in the series and on the right-hand
scale (inside) find the height at which it is to be plotted or (outside) the height
of chart-paper. The nearest circle on the central scale, to a straight line between
these two will give the side of the ruler (N) and the value of the ruler unit {U).
Comp, with Fig. 407.
558 CHJRTS AND GRAPHS
'/..T
/«-
%z-
X.-
X=-
X-
X.--
%-
On* Sixt««inih'
Thr«« Thirty- ••Condi-
On* Slghth'
Thr«» Siixteanths-
On» Gu«rtor-ton*'
Thr^e Bl)?hth8-|-
Cns H«lf-Ton*
three Cuarter-Tonea
One Ion*
thrae
two Whole Tonaa
thre* Whola Tonea"*"
T4
IS-
IS
11
10
9-
fi
7-
6'
I
Z-
Dift*na« of
Kel« from
tru* C*ivt*r
«f Spiral 0roo<r**
(In inch**)
Vkrlatlona
of Pitch
of Music
(in ton*s)
Dittnnc* of
N«odl« from
CtnUr of Di»e
(Tru* Cantor
of Spiral Oroo***)
(In inchoa)
KPrect OF OFF-CWTER HOLES IN PHOHOIRaPH RECORDS ON PITCH OF MUSIC
Figf. 432. Parallel Nomograph Not Chartable by Formula.
The unknown variable, T ( — change in musical pitch, in tones of the chromatic
scale) cannot be plotted by the side of a slide-rule, or any other ruler. The formula
of the chart is,
s= 0
R
in which R is the radius or distance of the needle from the true center of the disc
(in inches) and D is the displacement of the hole therefrom (in inches). The
formula can be stated as
ID
= 2 - 1
or log (2D) - log J? = log (2 - 1)
or log D — log /; = log {1^'^ — 1; — .3010
If we lay off logarithmic scales of D and R through the desired ranges and then
compute each value of these for each value of T which it is desired to plot, we
can plot same by isopleths through the computed values of D and R. Thus:
T
log 2
T
6
or .05017 T
log 2
2 1
R
or 2D
D
6
or 2
set
.05017 a
antiiog b
c — I
set
cle
VA
a
b
c
1 ^
f
g
1
.05017 1
1.1225
1225
5.
.6125
.30625
2
10034
1 2599
.2599
5.
1.2995
' .64975
Vz
.0250H
1 05946
.05946
5.
2973
‘ . 14865
etc. j
etc.
etc.
etc.
etc.
etc.
etc.
PARALLEL NOMOGRAPHS
559
can be plotted if it can be reduced to the additive form qx +
P%=^{p+q)y-
The great difference between the nomograph and the
curves described in the previous chapter, is that the curve of
the latter has shrunk to a point in the nomograph, and the
succession of curves has shrunk to a succession of points or
single line. Incidentally the nomograph has shaken off the
network of co-ordinates, though these are nor essential even
to the curve. In both of these steps the nomograph has re-
duced the labor of chart-making and increased the ease and
accuracy of chart-reading. We have so far considered only
the simple parallel nomograph, which is analogous to the
simple parallel curves, but there are other nomographs which
serve the purposes of the radiating curves and composite curve
charts which we shall take up in the next chapter.
Chapter XLVII
ZIGZAG AND COMPOSITE NOMOGRAPHS
In a curious way the zigzag form of nomograph is even
simpler than the parallel form. The parallel form has two
outer axes and an inner axis which is slid along the base-line
back and forth between the two outer axes as the scale moduli
or coefficients of the variables on the outer axes are changed.
In the zigzag form this central axis shrinks to a point — its
own zero-point or origin on the base-line, — and having so
dwindled moves back and forth along that base-line as a
variable along a scale. It shrinks to a point because the third
variable is turned into a constant. It lies upon the base-line
because the new constant has been corrected for on an outer
axis and one of them plotted reciprocally, that is, downward
from the base-line. It moves back and forth along the base-
line because the coefficient of the remaining independent vari-
able has been turned into a new variable and hence has vari-
able values. As a result, we must calibrate the base-line itself
for the values of this new variable coefficient, or factorial vari-
able, and lo and behold, we have a factorial chart without
log projections, in many ways similar to the factorial radiating
curve-chart for calculating formulae.
It- is simplest, however, to explain the zigzag nomograph
independently and from a different form of the geometrical
theorem of similar triangles. We shall now speak of the base-
line as an axis in itself, since it is calibrated, but it is to be
understood that distances are not measured off upon it in units
necessarily commensurable with the distances upon the other
axes. But we anticipate.
If you lay off three lines or axes such that two are parallel
and the third cuts through both, like the letter N, you can
easily prove that along the three axes the distances cut off
by a straight intersecting cross-line (in the finished chart, an
isopleth) will have certain definite relations. In this case
560
ZIGZAG AND COMPOSITE NOMOGRAPHS 561
we measure the distances from the intersections of the axes,
that is, the two intersection points of the axes, are the origins
of the axes. Let us call the first axis as before the x-axis and
the distance laid oiF on it by the isopleth from the intersection
of (x) and (y), as before, x. Let us call the middle or diagonal
axis, the y-axis, measuring the distance, y, laid off on it by the
isopleth, from the intersection of (x) and (y). Let us call the
third axis, as before, the s-axis, measuring the distance, s, laid
off* by the isopleth, from the origin of the s-axis, that is, the
intersection of the y and z axes. Now if we indicate the entire
length of the y-axis, from x-origin to 2;-origin by Q to indicate
that it is constant, we can quickly, from similar triangles,
verify the following statement:
X ^ y
s ~Q -y
Obviously if (with a chart-modulus, w = l inch) we plot
the values x^X, % = Z, and^^-^= F, we may use this chart
X
for calculations of the equation Y or X = FZ. As in the
parallel nomograph, we note that space is conseiwed and accu-
562
CHARTS AND GRAPHS
racy gained by plotting the dependent variable upon the
central or (y) axis.
AX + R
The generalized equation is Y = \-£‘
CZ ^,‘ + D
Lz
As in the last chapter, we concern ourselves first with the
two outer scales. Their ranges are:
= Rz=Hz-Lz
If we determine upon the height of the chart, Tm, or T inches,
when the chart-modulus is 1 inch, then the scale-moduli are
found from the two measures of the length of the scales
Rx'^x =Tm Rz'^h =Tm
or, letting w = 1,
Tm
These are sufficient plotting instructions for the two outer
scales if we are using a radiating scale-sheet.i But if we are
using an engineer’s rule, we shall want them turned into
values of S (the number of scale-moduli per inch) :
and if S does not at once show a ruler-copying value, that is,
1, 2, 3, 4, 5, 6, or decimal multiple or submultiple thereof, we
shall alter R and T a bit until it does for each scale.
Now we can at once lay off the central or y-axis. Care
must be taken to direct it at the true origins of the outer axes,
which will differ from the apparent origins by the amount of
JB £)
^(on the x-axis) and — (on the z-axis). The entire chart
should be about square in outline (unlike the parallel nomo“
graph) .or even slightly narrower, to get the best results in
^ That is, a sheet facilitating scale adjustment by triangulation. These sheers
arc found in Professor Lipka^s book and are described in the previous chapter.
ZIGZAG AND COMPOSITE NOMOGRAPHS 56 ;,
reading from its isopleths. The scale for Y can be inserted
by solving the equation for different values of the variables
and plotting them on the y«axis by isopleths. The scale is
not uniform, it is a variety of reciprocal projection, each
point having a value proportional to the ratio of the segments
of the line on either side of the point. Every value calibrated
on the scale must therefore be individually computed and
plotted by isopleths. This is a simple way to construct a
zigzag nomograph.
Since, however, the projections through these points or
values from any point, on the s-scale, forms as we know,
an arithmetical scale on the ;c:-axis, it is a simple matter to
reverse the process and plot a temporary working scale (which
we may call W) arithmetically along this A;-axis, calibrated
equal to Y and such that from it we may easily project the
T-scale on the y-axis.^ We select any convenient point, iiy
preferably near the middle, on the 2;-axis. Through two known
points already computed and plotted on the central scale, w’e
project isopleths from n to the x-2ixis. Then we lay off a
complete scale about these two points and with n as center,
plot their projections on the y-scale; lastly we erase the tem-
porary scale and the point n. This is a better way to con-
struct a zigzag nomograph.
The student will seek a mathematical expression for the
plotting of this eccentric y-scale. Now to find the values of y
m X 'V
in the equation — = --7^ , we must first examine the true
^ Q-y
values of x and %. These are the distances along the two axes
from the true origins, which differ from the apparent origins
by the amounts of ^ and
. A G
write
B
So to be quite correct -w e must
% = m,XZ+~)
I'he projections of )' upon the v\*-axis from any point in the s-axis are always
regular uniform or arithmetical, or in accordance with the scale of A'), because
A' itself is regularly laid off, and by the formula, Y varies directly with A" when Z
is taken constant. The temporary working-scale, Wy cannot be plotted upon the
li-axis from a point on the y-axis, because, by the formula, Y varies inversely with Z
when A" is taken constant; hence, the transversals from uniform intervals along the
z-axis would only project upon the y-axis a sort of reciprocal scale projection thereof.
564
CHARTS AND GRAPHS
This is true from the definitions of and as scale moduli.
So
x^'^iAX+B) z=^{CZ+D)
t X V
Substituting these in the equation — =7^ > have
y _ CnixiAX +B)
Q -y Am^{CZ+D)
Now
AXPB
CZ+D
= Y -E, so we may write
y ^ Cm^{Y -E)
Q -y 'Am^ Am^
Q -y _Q _i
y y Cm,iY-E)
Q Cnix{Y -E)+Ami
y~ Cm,{Y-E)
y =
Cm^(Y-E)
Cm,{Y -E)+Am,^
This is a cumbersome expression and shows that it is simpler
to compute the y-scale empirically as before described. If we
write
y—my(Y -E)
then we see that my, the modulus for the y-scale is
my(J-E)
Cm,{Y-E) Q
Cmx{Y -E) -pAm,,
yy, Cm,,Q
• Yyi ss '
^ Cm^{Y -E) +Am,
from which we see that this modulus is not constant but
changes with the values of Y, the variable. This is merely
•algebraic proof of the irregular nature of the y-scale projection.
We may observe from the equation for y that it is a fraction
ZIGZAG AND COMPOSITE NOMOGRAPHS 565
of the constant distance, Qy between the true origins of the
outer scales and that y (and hence the scale for the Y variable)
will always lie between these origins (and hence between the
two outer scales) so long as the denominator of the fraction
exceeds the numerator; we likewise observe that the y-scale
will lie outside of the two parallel scales when the numerator
exceeds the denominator.
An interesting thing about the y-scale is its F-value or
calibration at the point midway between the two outer scales,
that is, when y = | 0 . For this we write
, Cm^{Y-E)Q
^ Cm^{Y-E)+Am^
CmJiY -E) +Am, = 2Cm^{Y -E)
Ainz = Cmx{Y -E)
P Am^
~ Cm„
y- Anig
Cm^ '
E
YE
Thus at the point midway between the two scales the value of
Y is always easily found, by dividing the constant coefficient
of each independent variable by its scale modulus, then divid-
ing the quotient for x by that for z and adding any added
constant. If we have the simple case in which both inde-
pendent scales were plotted on the same moduli, and there is
no added constant, then the midpoint of the (y) scale expresses
the ratio of the coeflGcients in the formula. If the coefficients
are alike but the moduli are different, and there is no added
constant then it expresses the ratio of the moduli. Of course
the addition of a constant, E, merely raises these values by
its amount. And when coefficients and moduli are alike, and
no constant is added, the midpoint has the value of 1, and at
equal distances on either side all the other calibrations will be
found to be mutually reciprocal.
Useless as is the mathematical expression for y and niy, a
similar expression for the temporary projection of the (y)
S66
CHARTS AND GRAPHS
scale upon the x-zxiSy by means of which the y-scale can be plot-
ted without computing, is valuable. We select on the js-scale
(preferably near its center) a fixed point, n, the calibration
or Z-value of which let us call N. If we run transversals
through N and every uniform value of Y to be calibrated on
the y-scale, we know that the transversals would mark ofF a
regular scale upon the x-zxis. Let us call the temporary work-
ing scale zo. The intervals or scale modulus, of the zv-
scale, would have the same relation to the modulus of at as
the calibrations of X have to the calibrations of W. Thus
^ Now when Z=Nj the value of ^ is as follows
W Y Y
AX+B = {CN+D) (Y^E)
X = CN+D j^
A A A
Drop the added constants since they merely shift the zero
point and
X — CN -\-D y
Y
Mx
CN+D_X
A W
CN+D
A
CN+D
A
-nix
Or if we wish to work with the length, n, of the plotted point,
from the true s-axis origin, instead of its calibrated Z-value,
Ny we have
n = {N+p=^{CN+D)
C ^
A
n.
If we wish the modulus in terms of the ruler to use, i.e., the
number of moduli per inch (or per chart-modulus, whatever it
be), we have
c —
" CN-+D CS, '
® The mathematical steps here are outlined without full details, as the latter would
make the equations more cumbersome than their importance justifies.
ZIGZAG AND COMPOSITE NOMOGRAPHS 567
In short, to prepare a zigzag nomograph for the equation
JX
L.
cz
H,
L,
+D
+E,
we need only compute the following expressions in order to
plot with an engineer’s hexagonal rule:
in which approximates R^=IH
in which approximates -L.
CS,
in which n is the distance (in inches or units of the
n chart modulus) of a fixed point on the %-axis from
the origin thereof,
or
AS,
CN+D
in which N is calibrated Z-value on the s-scale.
The usefulness of the above expressions is in the search
for ruler-copying values of S which will enable us to plot di-
rectly from the engineer’s rule. For this purpose the same
tabular arrangement of columns for the values of r,, S,, AS,,
n, S^y CSsjSs, r„y and should be made in order that slightly
different values of t and r may be tried on various scales.
If, however, we work with a radiating scale-sheet, then the
scale-moduli are wanted and these are as follows:
nix
niu
or
L
'R.
JL
"r^
Am.
CN+D
n
The chart will also express the exponential equation
BX^ = ^ ^ = log 7+log E
^ CZ+D ^
and amplifications thereof; here one outer scale is logarith-
mically, the other arithmetically projected. When the other
CHARTS AND GRAPHS
568
functions of the variables are used, such as powers, roots, or
reciprocals, the corresponding projections may be called for.
In all this work the chart will be seen to handle added con-
stants with much less trouble than the factorial or logarithmic
Fig. 434. The T-Scale Outside the Parallel Scales.
Plotted with = —2 (the negative sign shows that scale-values increase upward
along this axis, instead of downward as normally), Sz — 6, and Sw = .01 for
33 §. The positive sign of Sw shows that /F'-values (on the JT-axis) increase (that
is, become larger positive or smaller negative values) downward, as normally; this
consequently applies also to the T-values on the F-axis. The upward or reversed
projection of the X-scale (due to the negative sign of shows that the F-axis,
passing through the true zeros (plotting origins) of the parallel axes, lies below
and outside the two scales, hence the F-scale is outside the parallel scales.
This form is not of much value. The dependent variable would better be X
when it is used.
(Note: Above dimensions as of original drawing, here reduced to half-size.)
parallel nomograph, for it does not require a specially com-
puted scale for the shifted zeros. The zigzag nomograph is,
however, on the whole of less value than the parallel nomo-
graph, for the central axis, being diagonal, is often crossed by
the isopleths at very small angles, and the readings naturally
become less accurate.
ZIGZAG AND COMPOSITE NOMOGRAPHS 569
Many other forms of nomographs have been devised beside
the parallel and zigzag forms. The theory and making of these
involve mathematical \vork and detail outside the scope of this
book. They are based upon various geometrical theorems
Fig. 43S. The F-Scale Inside the Parallel Scales.
The same equation as in Fig. 434, here plotted with Sx-2, Si-6y and S%v
= — 0.01 for V==33 J-. Ihe positive signs of S.v and Sz show that the normal
directions of plotting of the X and Z scales obtain, X increasing downward and
Z upward and the Y axis consequently passing between them. The negative
sign of Sw shows that the /F-values along the A-axis and therefore the I’-values
along T~axis, increase upward instead of downward (that is, grow larger positively
or smaller negatively).
This form is usually better than that in Fig. 434, for though it gives less
detail to parts of the F-scale, it places the dependent variable inside, giving
more accurate readings.
(Note: Reduction of half-size.)
and are generally built up with straight lines for scales on the
sides of imaginary triangles and parallelograms. It is indeed
possible to have nomographs with curved axes but these are
not often encountered, nor is their need more than exceptional.
A very large body of the less used nomographs are propor-
tional, and can be used for equations containing four variables
which are in or 'can be put into the form of a proportion.
These charts use two isopleths either parallel, perpendicular,
or with intersections upon a dummy line, in order to afford
the readings for the four variables. The interest attaching
to these less common types of nomographs is still largely
570
CHARTS AND GRAPHS
academic; the two simple forms, the parallel and zigzag, afford
adequate calculating facilities for all practical purposes.
We have so far considered only the simple forms of these
nomographs, in which there appear but a single set of three
axes, and which are suitable only for equations with three
variables (two unknown). The most interesting form of nomo-
graph is a compound one composed of two or more inter-
locking single nomographs and suitable for equations with
more than two independent variables. Each single nomo-
graph is a set of three axes, but when two or more sets are
combined, one of the axes of each s^t serves double duty,
Fig, 436. Construction of Factorial Zigzag Nomograph — Unfinished.
ZIGZAG AND COMPOSITE NOMOGRAPHS 571
Note to Fig. 436
Showing the working-scale, JF, from which the dependent variable scale, F, is
plotted; and the true origins, of the independent variable scales, shifted for added
constants.
The equation is 5 a® (5.7171^)^+^
in which a varies from 1 to 100, c varies from 3 to 13, and b is to be found. Let
X = a, Z = Ct and Y = b.
Since log 5 + 3 log <2 ~ (c + 2) (log 5.7171 log b)
3 log a + log 5 , , , , p
or log & log 5.7171
and the typical formula is
AlogX
IIx
Lx
+ log B
CZ
Hz
Lt
+ D
log F — 'og
we have, by substitution
3 log X
log 100
log 1
13
+ log 5
^ log F- ( i
+ 2
log 5.7171)
If wc wish the chart to have a total height of F =*5 inches we can plot with the
following scale moduli:
IJx - Lx - log 100 - log 1 = 2 - 0 = 2 Hz - I:: - 15 - 3 = 10
T
vix =
5
= — =2.5 inches
mz =
T
5
— = .5 inches
Rx
2
Rz
10
We must plot the :v-scale from a logarithmic scale having one deck for every 234
inches, and the s-scale from the 20-side of an engineer’s ruler, after allowing for
the added constant in each case.
If we select as the fixed point, iV, for our working scale, W, the point calibrated
as 10 on the z-scale
CN-^D 10+2
— ^2.5) = 10 inches
A 3
and we plot the 52;“scale from an inch-rule (the 10-side of the engineer’s ruler).
To position the /F-scale we calculate the value of X for any value of I" we choose:
thus,
when Z = W = 10 1 , ^ „ (Z + 2) (log F + log 5.7171) - log 5
and F(= r) = 11 3 ^
12 (log 1 + .75722) - .69897
3
= 2.7959
X = 625.0
While this value of A” lies outside the range, and is therefore inconvenient, we
need not recompute A" for another value of F, but merely extend the A" scale
sufficiently to plot JV = 1, after which other W values follow by the ruler.
CHARTS AND GRAPHS
ST-
being common to two sets and effecting the combination.
Thus if we have the formula or equation A = B-\-C-\-D+E,
we will have to break the right side of the equation, having
6a® • (6.7i71b)(c+ 2)
Fig. 437. Construction of Factorial Zigzag Noraograph — Finished.
The temporary working scale, /F, is erased after the calibration of the y-scalc
therefrom. Also the extensions of scales to the true origins and to x — 625 have
been erased. Scales have been calibrated on both sides to facilitate readings
when using an opaque ruler as isopleth.
four independent variables, into two groups of two each and
make a parallel additive nomograph of each group, adding a
third axis to each to express the resultant of each group, and
then we can combine the resultants in a third parallel nomo-
graph to show the dependent variable, A. We would write
/ = 5+,C
and g^D+E
and A^j+g
In the first two groups w^e might let / and g be middle axes,
but in the third group we would use them as outer axes writing
A as the middle axis of the group. The order of the axes
would be 5, /, C, Ay D, g, E. If for convenience we wished
A to be the final axis, then we should have to fall back on the
use of inverted scales and carefully arrange the scales so that
a.5-T-2,S
ZIGZAG AND COMPOSITE NOMOGRAPHS 573
Fig. 438 is a compound nomographic chart by means of which parallel nomo-
graphs may be constructed.
Draw isopleths from Rx and Rz on the R scale (first) to Tx and Tz on the T scale
(fourth) and find scale moduli, mx and mz and rulers Sx and Sz on the Sm scale
(second). From the latter draw isopleths to A and C on the fifth scale and read
the values of p and q on the third scale. Add the latter, p and to get Sy. In
the above Rx and Rz are the ranges of the two independent variables, x and z;
Tx and Tz are the tenths (in inches) to be given these scales on the chart; Sx and
Sz are the engineer's rules (or number of units per inch) to use in plotting them,
A and C are the coefficients <jf the two variables, x and y; and p and q are the
horizontal distances between axes {p between x and y, q between y and z), Sy is
the engineer's mle (or number of units per inch) to use in plotting the dependent
y-scale. Position the scales (for added constants) by a single trial isopleths.
Fig, 438, Chart to Construct Parallel Nomographs.
574
CHARTS AND GRAPHS
Fig:. 439. Chart to Construct Zigzag Nomographs.
Here is a compound nomographic chart by means of which zigzag nomographs
may be constructed. Draw isopleths from Rs and Rz on the first scale to Tx and
Tm on the third scale, and read Sx and Sz on the second scale. From the latter
draw isopleths to A and C on the fourth scale and note intersected points on the
dummy axis. From the last, the intersection of the dummy scale and the isopleth
through Sz and C, draw an isopleth to n on the fifth scale. From the other
dummy axis intersection, A Sx, draw a parallel isopleth to the fifth scale and read
S^, In the above A and C are the coefficients of the independent variables, x
and z; Rx and Rz are their ranges; Tx and Tz their scale lengths in inches; n is
the fixed point distance on the z-scale to project the working scale, JF, on to the
y-axis as Y, and Sx, St, and Sw are the sides of the engineer's rule (or number of
units per inch) to use in plotting. Position the scales for added constants by
means of these plotting-units.
ZIGZAG AND COMPOSITE NOMOGRAPHS 575
while each y and z scale went in the same direction, each
scale went in the opposite one. Compound nomographs can
be used for equations" with many factors instead of terms, in
precisely the same way, merely using logarithmic projection or
zigzag nomographic form. The sub-total axes (/ and g in the
example just cited) or the sub-product axes in factorial nomo-
graphs, are generally left without calibrations, as no one is
interested in reading their values. They are necessary merely
as fixation points secured by the first interpolation and fixing
the isopleth for the next step. They are called dummy axes.
The fact that the zigzag nomograph performs multiplica-
tion and addition on arithmetically projected scales makes it
useful for compound nomographs of formulae involving both
addition and multiplication. This, indeed, is the chief reason
for the importance of the zigzag form. Thus an equation of
the general type A = BC +- DE can be solved by the use of two
zigzags for the two multiplication processes and a parallel
for the sum of their products. This equation could not be
shown on parallel nomographs alone, because in them logarith-
mic projections would have been necessary for the factorial
processes and the addition of the products, were logarith-
mically projected would have shown not a sum but a third
product.
It has already been said that many other projections can be
used beside logarithmic and arithmetic ones. Squares, cubes,
roots, and trigonometric functions can be used. When such
functions are used, the equations px == mX ox x = m^X no
longer hold, but must be modified to px = mf (X) and x =
m^f {X). This will require the modification of the calculating
formulae which have been given for the scale-moduli, but the
procedure is so similar that it may be left to the devices and
ingenuity of the chart-maker. Nomograph-making presup-
poses a fairly thorough understanding of the equation to be
plotted and with this as a basis, the ingenious experimenter
will find various and adequate methods of charting.
Upon the finished chart the scales should be provided with
titles below or above them, explicitly stating the variables to be
located or read on each scale. The formula which the chart
expresses should also be available to the reader somewhere
about the chart. The best mechanism for the reading of the
scales is a strip of transparent celluloid with a fine straight
576
CHARTS AND GRAPHS
line drawn in ink upon its lower surface. A straight-edge or
ruler, if possible with a transparent edge, can be used; and in
an emergency a piece of thread can be .drawn tight and held
for the readings.
Fig. 440 . In Quadratic and Cubic Equations the Position of the Central
Axes Becomes Variable, and a Chart-field Takes the Place of a Single
Scale.
This is the Darville-Johnson Bond-Yield Chart, a chart for determining quickly
the yields of all types of bonds, including premium bonds, maturing in any num-
ber of years at practically all coupon rates now in use, including odd fractions.
— Published by Prentice- Hall, Inc,
Whole books have been written about the nomograph and
while it is still a little known chart, yet it is fast increasing
in popularity, and deservedly so. It is the most easily con-
structed and accurately read of all calculating charts, and the
results which can be accomplished with it are always a source
of amazement to the uninitiated.
Chapter XLVIII
SLIDE RULES
A calculating device which is even more simple to operate
than the alignment chart or nomograph, but may be considered
closely related to it, is the slide-rule. We have seen that by
the use of special projections, the curve of an equation may
often be straightened, increasing both the ease and accuracy
of calculations based upon the curve. We have seen that in
the nomograph. the chart field has also been eliminated with
still further simplicity and benefit. But we come now to the
slide-rule, in which even the straight-edge (that rudimentary
substitute for the curve) has been eliminated, the scales losing
their fixed position with regard to each other and being freely
movable. The slide-rule is therefore nothing more than two
movable scales along the same axis, that is, in contact with
each other.
We can, however, approach the subject of slide-rules even
more simply if we consider first the stationary rule. The
stationary rule is nothing more than a single axis bearing two
or even more scales, one upon each side of the axis. This is
the graphic chart of equations containing but two variables,
(only one independent or known variable). The scales are so
adjusted that for every value of one variable, calibrated on
one scale the corresponding value of the other variable may be
read upon the other scale at precisely the same point along the
axis. Fixed or stationary scales may be laid off. upon logarith-
mic, arithmetical or any other projections or combinations of
projections. Such charts are useful in the place of small con-
version tables, but must be made large or in several segments
when detailed readings are necessary. They can be used as
ready reckoners for foreign exchange, temperature equivalents
in Fahrenheit and Centigrade, the conversion of metric and
common systems of measures and weights, and an infinite
577
578
CHARTS AND GRAPHS
variety of similar cases in which the relation between two
variables is constant and fixed.
V«Xoolty rorc«
per
As its name suggests, the slide-rule is not
fixed or stationary. If you will take two
ordinary rulers, one with its scale upon the
upper edge, and the other with its scale upon
its lower edge, and bring them together so
that the two calibrated edges will fit together,
you will have the simplest form of slide-rule.
Calculating is done by the simple trick of
sliding one ruler along the other and reading
the corresponding values in the new positions.
Thus in order to add 2 and 4, you need only
slide the upper rule along the lower one, until
the zero point on the upper rule is over the
figure 4 on the lower rule, and then read the
figure on the lower rule below the figure 2
on the upper rule. Obviously you have in
this way added two inches to the four inches
on the lower rule and you will get six inches
on the lower rule. The upper rule merely
tells you how much you have added to the
original distance on the lower rule. Likewise
to subtract 2 from 6 you need merely place
the 2 on the upper rule over the 6 on the
lower rule and read back to the figure 4 on
the lower rule under the 0 on the upper rule.
This amounts to deducting 2 inches from
the original 6 inches on the lower rule, giving
you a remainder of 4 inches on the lower rule.
In short, the slide-rule is merely a device
for the direct addition or subtraction of dis-
tances.
In the device just explained, the calibra-
tion of the two rulers forms an arithmetical
series and hence the calculating power of
this device is limited to the processes of ad-
dition and subtraction. In order to use the
fBB FOftCB OF mm
(StAXkdATd Table)
Fig^. 441.
A Stationary or
Fixed Rule.
device for the processes of multiplication and
division, of course it is only necessary to
calibrate the rules upon logarithmic projec-
tions, so that the addition or subtraction of
SLIDE RULES
579
the logarithmic distances will indicate the processes of multi-
plication and division of the numbers appearing on the scale.
As in the case of the nomograph, the calibration can be an
arithmetic or logarithmic projection of sine, tangent, square,
cube, root, and other functions of numbers as well as of the
numbers themselves.
The ordinary commercial slide-rule is nothing more than
a series of these scales of logarithmic projections of vai'ious
functions, mounted upon bits of wood which fit closely to-
gether and can be conveniently handled. One rule is made
Courtesy of Keuffel Esser, N. Y.
Fig. 442. A Slide Rule.
For multiplication of numbers, squares, cubes, tangents, sines and other circular
functions and also showing logarithms of numbers.
much smaller and fits within a groove on the other, sliding
freely back and forth along that groove so that it is not neces-
sary to hold the two rules constantly together. They are so
tightly adjusted that the two rules remain without shifting in
whatever positions they are placed, leaving you free to take
the readings on the scales with great care. The inner rule is
called the ^‘slide.’’ A ‘Tunner’’ is also attached to the outer
rule for convenience in taking readings, being generally a small
piece of glass on which a fine hair-line has been drawn at right
angles to the scale. When you have positioned this runner so
that the hair-line crosses the point desired upon one scale,
the hair-line will also cross the desired point upon the
other scale, and the reading on the second scale can be more
Permission of Keuffel Esser ^ N» Y.
Fig. 443. The Magnifiers Increase the Accuracy of Readings.
580
CHARTS AND GRAPHS
< w CO u
g O
Si ’T?
^ c
taJO
C
CQ
I 'X^
I o
s
0 )
cl
SLIDE RULES
581
easily taken. And when the two scales which you are using
are not in immediate contact, but are parallel some distance
from each other, theVunner is necessary to project the desired
point from the first scale to the second, forming a sort of
ordinate across the two scales. Magnifying glasses are often
attached to these runners so as to facilitate more exact readings.
Because the construction of a straight slide-rule calls for
rather delicate carpentry ,1 slide-rules on short notice in the
home or office are more easily made in circular form. And
because a circle is endless and over three times as long as its
diameter, the circular slide-rule can be made on much larger
scale and with consequently greater accuracy than a straight
slide-rule of the same physical size. For the circular slide-
rule, you need merely pin together through their centers two
Courtesy of Keuffd JEsser, N, Y.
Fig. 445. A Circular Slide Rule — Pocket Size.
Owing to the great length of a circumference (compared to a diameter), and to
the overlapping (because endless) edges, the circular rule is very compact.
circular pieces of paper, the smaller one uppermost, so that a
scale can be drawn on the visible inner edge of the lower one
which will always be in contact with a scale drawn on the
outer edge of the upper disc. Then by rotating one disc above
the other the readings can be taken off in precisely the same
way as with a straight slide-rule in which one rule was slid
along the other. The circular slide-rule operates on precisely
the same principles as the straight slide-rule, adding or sub-
^ Excellent examples of straight slide-rules for special purposes may be found in
the writings of Mr. Carl Barth, The circular slide-rule has been more used by Mr.
Walter N. Polakov.
582 CHARTS AND GRAPHS
ti'acting distances around a circumference instead of along a
straight line, the distances on the scales being prepared by
angular instead of linear measurement.
Fig. 446 . A Special Circular Slide-Rule.
Devised by Mr. Walter N. Polakov . — Fermissmt of Mr. Polakov,
Some difficulty may be met in the calibration of the cir-
cular scale. It is comparatively easy to project any scale or
calibration upon a straight line, but to project it upon a cir-
cular line or upon an arc of a circle, it is necessary to use an
ins'trument for measuring angles, called a protractor. Pro-
tractors are almost invariably calibrated in degrees, the entire
circle being divided into 360 degrees. Two other units of
angular measurement are known, grades (6400 grades to the
circle) and radians (the radian being the arc equal to the
radius of the circle), but neither of these is any more useful for
SLIDE RULES
583
the purpose in hand than the degree. Even the metric system
has no decimal unit of circular measurement. This is unfor-
tunate because a decimal circular measurement system would
often be convenient. Sometimes circles are divided into one
hundred parts for the plotting of 100% circles or pie-charts
but these are not usually of sufficient accuracy and precision
to use as protractors. Your best plan in the making of circular
scales is to decide beforehand approximately how far around
the circle you wish your scale to run and then turn your scale-
distances which you would use in calibrating the straight line
scale into the nearest convenient number of degrees and lay
them off with a large protractor or scale of degrees, last of all
re-calibrating the scale for the desired value from your con-
version table. Thus, if you wish a scale which runs from 0 to
10 in actual distances to extend about a quarter of the way
around the circle, you can plot your table of distances directly
onto the circle from your protractor b}^ using that portion of
the protractor which extends from 0 degrees to 100 degrees,
but if you wish your scale to extend over half way around the
circle you must first double the actual distance values before
plotting them as degrees, so that you can plot through the
protractor from 0 degrees to 200 degrees.
Circular slide-rules can be made with a number of inde-
pendent scales, each sliding on separate pieces of paper but
all pivotted together at their centres by a small rivet. A sub-
stitute for the runner can be attached in the form of a strip
of transparent celluloid with a fine ink line drawn radially
from the center or pivotal point. This ray can then be swung
about the circle and laid over any desired point on the scale
to facilitate readings in the same way as the runners on a
straight slide-rule. Another device is to make the uppermost
circular sheet of paper so large as to cover all the other sheets
and then cut windows or circular slits in the upper sheet where
the lower scale should be seen and mark small pointers or
arrowheads next to the windows for readings on the lower
scale,
Circular slide-rules are easily made when you have once
grasped the fundamentals of their construction and they afford
the greatest play for ingenuity. When skilfully made, they
perform the most intricate calculations with astonishing ease
and simplicity. Their only limitation appears to be that they
perform in one operation only that particular type of mathe-
584
CHARTS AND GRAPHS
matical process for which they have been designed. As they
operate by adding and subtracting distances, they will perform
Fig. 447. A Circular Slide Rule with Many Variables.
Showing Cost of Book Printing.
addition and subtraction if the calibration is in terms of
arithmetical series, that is, in the natural numbers. They will
perform multiplication and division of as many factors as
there are scales, by the simple trick of calibrating them log-
arithmically, that is for the logarithms of the natural numbers.
But they cannot be used at the same time for addition and
multiplication, for the two processes require different types of
calibration or projection of scales. This limitation is ordin-
arily not a serious one, because most tedious business compu-
SLIDE RULES 585^
tations are either processes of addition or multiplication but
not combinations of the two.^
Fig. 448. The Same as the Preceding, Except that All Scales are
Covered and Seen Only Through Small Open Slots or Windows.
When facilities are at hand for delicate and precise car-
pentry work, the straight slide-rule principle can be elaborated
and developed by a series of pulley wheels with cords passing
over them and connecting movable pointers along separate
fixed scales. The pointers can then be adjusted for the par-
^The scale-moduli of slide-rules vary inversely with the coefficients (in additive
rules) and exponents (in factorial, or log, rules) of the variables, and are alike as
these are alike. The length of the rules or scales, therefore, varies directly with the
ranges of variation of the variables (unless the moduli are unlike).
In the pulley-wheel and pointer type of slide-rule next described, the pulleys can
be made with various leverages and thus modify the moduli, affording opportunity
for the adaptation of the lengths of scales to any moduli, range, or length, desired.
SLIDE RULES
587
ticular readings of the variables and the final pointer will come
to rest at the correct reading of the answer for the equation,
the product or quotient of the various component factors. In
such devices it is even possible to combine the different proc-
esses of addition and multiplication in one machine at the
same operation, so that machines can be constructed to give
the answer for any set of variables in any equation. It is
hardly in the province of this book to go into the details of
construction of such calculating devices. The circular slide-
rules are, however, so easily constructed and sometimes such
great labor-savers and time-savers that it is well to be able to
make them, when occasion arises, especially adapted to your
own problem.^
3 It is by this time doubtless apparent that slide-rules and nomographs are clearly
akin. When we have an equation with one independent variable, we have a fixed and
rigid equality between it and the dependent variable, by which one is always a certain
function of the other. In such cases, the chart of the equation is a chart with fixed
or stationary scales. But when there are two independent variables, we can either
( 1 ) use sliding-scales, so that one of the variables can be eliminated by a proper setting,
or ( 2 ) use separated scales (that is, nomographs) so that one end of the isopleth can
be properly set to eliminate a variable. And over against these uses of scales, we have
always a more cumbersome alternative, either for one or two independent variables,
in the curve-chart.
Note TO Fig. 449
All wheels are fixed except the three in frames, which slide up and down. The
shaded bars are weights free to slide up or down, and sufficient to balance each
other, so that the string remains taut and at rest wherever it is set. The small
cross in the lower left-hand corner is the only fixed point to which any string is
attached. Computing is done by setting any pointers at proper points and
reading the answer on the remaining pointer.
Chapter XLIX
HUNDRED-PER-CENT TRIANGLES
Whenever you are dealing with problems in which three
elements or parts combine to form one whole, and you are
interested not in the whole but in the proportions of the three
parts, the computing of the problems as well as the recording
and presentation of the results can be accomplished with a
chart which we may call the 100% triangle. The chart is
also known as the trilinear chart and its rulings are sometimes
called areal co-ordinates. It differs from the 100% bar in
that it is limited to divisions of a total into three and only
y + * * A
Fig. 450.
three components, but it is similar to the 100% bar in that it
does not distinguish between large and small totals, all totals
J88
HUNDRED-PER-CENT TRIANGLES 589
being reduced to 100% and appearing upon the chart in exactly
the same size. It resembles the nomographic charts in that
it is a real labor-saver in the work of computing.
Like the nomograph, the 100 % triangle is based upon a
trigonometric principle. The theorem in this chart is that in
an equilateral triangle the sum of the three perpendiculars
dropped from a point within the triangle to the sides of the
triangle is constant and always equal to the altitude of the
triangle. The rule is limited to equilateral triangles, and the
100% triangle is therefore always made of equilaterals.^ The
three perpendiculars bisecting each side and extending from
the sides to the opposite angles of the triangle form the three
^‘axes” of the chart. As in the calculating charts, the word
^^axis"’ is here used in the special sense of a straight line to
which a scale is attached and from which ordinates are pro-
jected normally (that is, in perpendicular direction). Rulings
parallel to the sides serve to project the scales across the chart
in the manner of co-ordinates. A little thought will show
that the 100% triangle is merely a variety of the rectilinear
computing chart, in which the %-scale is calibrated so that
2;=:100 instead of z = x+y^ or that 100=A;+y+2;‘
From this it follows that the equilateral triangle is not essential;
any triangle can be used, but for reasons already pointed out^
the scales are most easily projected upon the equilateral form,
since then the distances along the three axes have the same
real significance.
The scales used for the axes of the chart may have anj?-
range, but the customary scale is an arithmetical one, cali-
brated in percentages and ranging from zero to one hundred
per cent. This forms the arithmetical 100% triangle. Data
to be charted upon it must first be turned into percentages of
the total of the three elements charted. The chart is an addi-
tive one, the three elements combining by addition to form a
total. As has been said, the chart does not distinguish between
the sizes of the totals, but shows them all of the same size.
Nor is it possible to attempt to show relative total sizes by
varying the dimensions of the equilateral triangle, as its area
^ Obviously, any other triangular form can be used, but the intersections of the
three sets of co-ordiriates become less sharp and well-defined, and the scale-moduli
become less easily commensurable. See Chapter XLIII.
2 See Chapter XLIII. The 100%-triangle is most fully described in Haskell,
Allan C., Horn to Make and Use Graphic Charts, Codex Book Co., New York.
590
CHARTS AND GRAPHS
increases by the square of the increase in its altitude. The
general form of the equation for the chart is *100% +y + 2 ;”.
The classical example of the use of the 100% triangle is'
the analysis of food values in terms of calories (or heat pro-
ducing units) of fats, proteins, and carbohydrates or hydro-
carbons (sugars and starches). The well-balanced ration
being about 20% proteids, 60% hydrocarbons and 20% fat, a
point can be located on a 100% triangle at the intersection of
these three ordinates and the approach of various foods and
combinations of foods to this ideal easily seen. Moreover,
the computing to plan a well-balanced meal can easily be done
upon the chart. Thus with the two points for bread and milk
plotted upon a chart, a line drawn between the two points
indicates all the possible food values which can be had by
mixing the two in various proportions. With equal caloric
amounts of each, the point midway upon this connecting line
will give the components of the combination. Foods known
to lie upon the opposite side of the ideal ration-point from
this mixture-point, must then be added to bring the meal
nearer to the ideal, a straight connecting line again serving
to show the results of combinations in all possible proportions.
Such calculations can be performed upon this chart in a frac-
tion of the time which they would require by any other method.
HUNDRED-PER-CENT TRIANGLES
591
In business this paper can be used in innumerable cases
vhere a total is divided into three parts. Advertizing appro-
A
Fig. 452. The Hundred-Percent Triangle for Food Values.
The cliart shows the fats, proteids, and carbohydrates (in calories) of chicken
bread and milk, the full lines show the results of mixing any two of these, and the
intersection of the broken lines shows the result of combinging an equal quantity
of each .- — Permission of Mr. Malcolm C. Rorty.
priations, for example, may be divided into magazine, news-
paper, and outdoor advertising- Salesmen are often expected
to preserve a certain proportion between their sales of large,
medium, and small profit lines. Inventories may be kept in
terms of raw materials, finished stock, and goods in process.
Assets are often divided into current, fixed, and intangible
assets; costs into payroll, materials, and overhead. In scien-
tific work the chart has been used for the chemical analysis
of mixtures of three elements. Extensiye use of the chart has
been made in engineering, for comparing various grades of
coal as to their hydrogen, oxygen, and carbon content; and
for investigating concrete mixtures and so on. In economics
CHARTS AND GRAPHS
592
the chart would seem admirably adapted to the study of
projects for the joint representation of owners, workers, and
Corapromlasd
^TTLEMENT OP STRIKES
Percentages of victories for Employers and Employeoa
and of Compromised Strikes
United States
1916*1931
(Arranged from Monthly Labor Review)
Figr. 453.
public in industrial disputes, or the division of earnings into
wages, salaries and dividends, or the division of authority
between management, workers, and stock-holders. The tri-
partite division is frequently called for.
By using logarithmic scales, the chart can be made fac-
torial instead of additive and used for cases where three
elements combine by multiplication to form a product. Its
equation is: log 1 = log :v+log y+ log z, or 100% = xyz.
As before, the chart will not show the size of the resulting
product, it will merely show the relative proportions of the
three components. The scale can be of as many logarithmic
decks as desired, according to the range of variation of the
components, but of course the same scale must be used for all
three axes. This chart is often better when its scales are re-
HUNDRED-PER-CENT TRIANGLES 593
calibrated to absolute quantities, the product of which is a
fixed given amount. The chart can then be used for the com-
xsz s 100
Fig. 454. The Factorial 100% Triangle.
parison of the component factors in two or more equally large
products. Its equation then is: log C=log ^r+log y+log z,
or C = xyz.
The logarithmic 100 % triangle has apparently never been
used, but it is almost as often desirable as the arithmetic one.
In engineering, for example, there are innumerable equations
involving three factorial variables. A most obvious case is
that of cubic measurements involving height, length, and
breadth. The commercial measure of electric current is the
kilowatt-hour, a product of voltage, amperage, and hours. ■ In
factory management, labor cost for a job is the resultant of
the number of workmen, their average hourly wage, and their
time on the job. In finance, business, and economics, similar
HUNDRED-PER-CENT TRIANGLES
595
equations will be met, involving three variable factors, and sus-
ceptible to analysis by this form of chart.
With a thorough understanding of the method of the chart,
it can also be used for division, by the use of reciprocals. The
equation then is log C = log ^r+log y -log s, or C = — . An
* X \
equation of the form C= — or C= , could equally well be
yz xyz
shown. Without logarithms, the chart thus performs sub-
traction, as 100% = X+r-Z or 100% =X-r-Z or 100%
= -Y -Y -Z. There is also another method for these cases,
which does not involve recalibration of scales. It is based
upon the more general geometric theorem that the algebraic
sum of the perpendiculars from the sides of an equilateral
triangle (or extension of the sides) to any point in the plane of
the triangle, is constant and equal to the altitude of the triangle
The method involves the use of the area outside of one of the
sides of the triangle, and hence, on the axis of that side, in the
negative part of the scale.
596
CHARTS AND GRAPHS
In common with the 100% bar, and the 100% circle or
pie-chart, the additive 100% triangle is equally significant as
2 Y X
X - y 4 z s 100
Fig. 458-
to the linear measurement and the area measurement of its
parts. For the plotted point within the triangle can be con-
nected with the three angles by three straight lines, and these
Goods lii Process
Fig. 459.
straight lines will be seen to break up the triangle into three
small triangles, whose areas are in the same proportion as
HUNDRED-PER-CENT TRIANGLES
S91
their altitudes, or distances along the scales of the triangle.
In this case, as in the 100% bar and pie-chart, the area of each
segment is significant because only one of its dimensions varies,
the other dimension being constant for the three segments or
areas. Obviously, the segmentation of the 100% triangle is
only useful in additive charts and only desirable for extremely
popular use, when the chart is to show but one set of data,
that is, one sum bi'oken into three parts.
For both additive and factorial equations, with either
absolute or relative values or units of measurement, the chart
will be found strikingly illuminating and an excellent means
of analysis and comparison within its somewhat narrow
limitations. For it must always be remembered that the chart
does not show totals, products, or other resultants. It shows
only the comparative sizes of the components. If, therefore,
it is the size of the resultant in which you are interested, the
chart is worthless, but if it is the proportions of its three parts
or factors, the chart is admirable, and is in fact, quite the
clearest possible means of analysis.
PART VI. TWO- AND THREE-DIMENSION DATA
Chapter L
HUNDRED-PER-CENT SQUARES
We have so consistently inveighed against the use of areas
to illustrate quantities that the reader will indeed be surprised
at some coming retractions, however guarded and limited
we may make them. But the fact is that we now propose to
turn to advantage the very feature of areas which has previ-
ously been their greatest fault. Let us examine this feature
closely and see how it can be done. The reader is, of course,
familiar with that elementary theorem of geometry and arith-
metic, which states that the number of square units of meas-
urement in an area is a product of the linear units of measure-
ment in its two dimensions. He therefore realizes that the
variation of both linear dimensions by a given ratio results
in the variation of the area itself by the square of this ratio.
This has been the dangerous stumbling block in the way of
using areas in charts which are intended to illustrate but a
single ratio or set of ratios. To prevent the confusion and
doubt which might arise in the minds of those who see our
charts, it has been necessary to maintain between the areas
the same ratios as exist between their linear measurements.
And to maintain this identity of ratios, we have been obliged
to keep one linear dimension constant wherever areas have
appeared. In the 100% bar, the 100% circle, the 100% tri-
angle, and in bar-charts, and even band charts, in short in
all charts using areas up to this point, we have invariably
striven to maintain one constant dimension, with this specific
purpose of making variations in areas equal to variations in
the one varying dimension.
We now come to data in which we wish to show simulta-
neously three ratios or sets of ratios, one of which is always
the product of the other two. In other words, we wish to
show two factors or sets of factors and their product. And
to this purpose it is obvious that the area is excellently adapted,
598
HUNDRED-PER-CENT SQUARES
599
by reason of the feature which has just been described. What
was previously an obstacle to the use of areas (varying along
both dimensions) now becomes an essential advantage. And
as in the case of bars (that is, areas varying in one dimension
only) so also in the case of areas proper (that is, areas varying
along both dimensions), we find that the charts can be di-
vided into two groups. The first group is composed of charts
in which the total or whole area is a constant and is cut into
segments whose sizes are of interest to us. The second group
is composed of a series of separate areas, which may or may
not be individually segmented, but in which the total sizes
of the individual areas vary. In the section on bars, the first
type was called the 100% bar; the second, a bar-chart. Simi-
larly in this section the first type is called the 100% square
or rectangle; the second, an area-chart or area-bar-chart.
Before proceeding to the separate consideration of these
two types, we must call attention to a general limitation hold-
ing for all types of area-charts. These charts, as we have said,
illustrate simultaneously two factors and their product, the
product being shown by the area itself, the factors by linear
dimensions. Now it has already been frequently pointed out
that the human eye cannot so easily or precisely judge of
square measures as of linear ones. Hence we must expect the
illustration of the factors to be clearer and more easily evalu-
ated by the reader than the illustration of the product. And
we may say in general, therefore, that the area charts are
desirable only for data in which the product itself is of less
importance than one or both of the factors. Where an ex-
plicit illustration of the products is necessary, afFording pre-
cise and detailed comparisons of the products, this chart does
not suffice, but where the primary importance attaches to
one or both of the factors, and the product is only of sec-
ondary importance, the chart will serve excellently.
Like the 100% bar, circle, or triangle, the 100% square
is a device for the illustration of the parts of a total. Unlike
them, however, the 100% square does not show only one
classification of the parts, but shows simultaneously two inde-
pendent classifications, which combine factorially to produce
a great many small parts. The data for the 100% square,
therefore, consists of two interlocking or mutually crossing and
subdividing classifications of the parts of a whole. Whatever
the absolute value of this whole may be, its relative value is
6oo
CHARTS AND GRAPHS
100%, and the chart is ordinarily calibrated in percentages.
If the data is not already in percentages, it can easily be
turned into percentages to facilitate charting. The important
Total
Proprietors
Managers
Officials
Clerks and
kindred
Workers
Skilled
Workers
Semi-
skilled
Workers
laborers
and
Servants
Total
41,609,192
11,165,536
5,638,144
4,914,651,
6,304,567
13,506,294
Agriculture, forestry and
10,951,074
5,601,742
5,449,332
animal husbandry
509,041
653,203
Extraction of minerals
1,090,854
28,610
—
—
Manufacturing and mechanical
12,812,701
660,622
—
4,669,126
4,247,232
3,215,721
1,530,704
Transportation.
3,066,305
206,352
455,305
225,525
648, *119
Trade
4,244,354
1,615,823
2,062,884
—
355,205
210,442
Public service (not
771,120
658,351
—
—
112,769
elsewhere classified)
23,677
Profeaalonal service
2,152,464
2,128,787
—
Domestic and personal service
3,400,365
365,249
—
—
600,993
2,434,m
Clerical occupations
3,119,965
I
3,119,955
•—
...
Fig. 460.
OCCUPATIONS OP THE GAI«FULL> tUFLOTED POPUtATlOR
(10 years of age and over)
The tnlted States
1920
(Source;- U. S. Bureau of Lah'r Statistics)
The Original Data for a 100% Square.
thing about the data is that it should be clearly arranged in
a tabulation or table, with the items of one classification listed
down an edge of the table as the stubs of the table, and the
Total
’roprletors
Managers
Officials
Clerks and
kindred
Workers
Skilled
Workers
Semi-
skilled
Workers
Laborers I
and
Servants
Total
100.00
26.84
13,54
11.83
15.33
32.46
Agriculture, forestry end
animal husbandry
26.31
13.22
...
...
13.09
Extraction of minerals
2.62
0.07
...
1.22
1.33
Manufacturing and mechanical
Industries
30.82
1.59
...
11.29
10.20
7.74
Transportation
7.37
.50
1.09
,54 '
1.56
3,68
Trade
10.20
3.88
4.96
— . 1
.85
,51
Public service (not
elsewhere classified)
1.85
1.58
...
...
...
.27
Professional service
5.18
5.12
...
—
,06
...
Domestic and personal service
8.16
,88
...
...
1,44
5.84
Clerical occupations
7*49
7.49
...
...
...
OCCUPATIONS OP THE GAINFULLY EMPLOYED POPULATION
(10 years of ago and over)
The United States
1920
(Source:- U. S. Bureau of Labor Statlstlesj
(percentages)
Fig. 461. In this Form the Data is not Chartable.
items of the other classification listed across the top of the
table as its column headings. In the body of the table, at the
HUNDRED-PER-CENT SQUARES 6oi
intersections of columns and rows,- are placed the detailed
figures which correspond to both classifications.
In turning the absolute detail figures into percentages of
the total, we have, of course, no trouble, merely dividing each
figure by the figure for the total. But in this form, the data
is no longer factorial, the detailed figures not being products
of factors which are known, and therefore not being amenable
to charting by areas. To draw areas we must know the factors
which will be plotted as the linear measures along the two
dimensions of the area. It is therefore of no use to us to turn
the absolute values into direct percentages of the grand total.
Instead, we turn the sub-totals for each row or column into
percentages of the grand total, and then turn the detail
Total
Proprietors
Managers
Officials
Clerks and
Skilled
Semi-
Laborer®
Vertically
Across
Workers
Workers
Servants
Total
100.0
100.0
26.84
13.54
11.83
15.33
32.46
Agriculture, forestry and
animal husbandry
26.31
100,0
50.3
...
...
...
49.7
Extraction of minerals
2.62
100,0
2.6
— .
46.7
50.7
Manufacturing and mechanical
Industries
30.82
100,0
5.2
...
36.6
33.1
25.1
Transportation
7.37
100.0
6.7
14.8
7.4
21.1 1
50.0
Trade
10.20
100.0
38.1
48.6
...
8.4 1
4.9
Public service (not
elsewhere classified)
1.85 !
100.0
85.4
...
—
...
14.6
Professional service •
5.18 1
100.0
98.9
...
...
1.1
Domestic and personal service
8.16 i
100.0
10.7
...
...
17.7
71.6
Clerical occupations
7.49
100.0
—
100.0
... !
...
OCCtJPATIOlIS OP THE gainfully EMPLOYED POPULATION
- <10 years of age and over)
The United States
1920
(Source;- U. S. Bureau of Labor Statistics)
(percentages)
Fig. 462. Here Each Row Totals 100%.
figures in the body of the table into percentages of the sub-
totals for the rows or columns in which the detail figures
occur. Thus we make the detail figures in themselves products,
that is, percentages of percentages.
In this step we come to a choice between turning the de-
tailed figures into percentages of the sub-totals for the columns
or into percentages of the sub-totals for the rows, in which
they occur. One or the other must be used, that is, either
the sum of the detail percentages in each column must add
up to 100%, or the sum of the detail percentages in each row
must add up to 100%. We cannot expect that addition up
and down by columns, and addition across by rows, will both
6oa
CHARTS AND GRAPHS
give 100% in the same table. We must, therefore, make a
distinction between the primary classification, in which the
sub-totals are percentages of the grand total, and the sec-
Total
Proprietors
Managers
Officials
Clerks and
kind'^ed
Workers
Skilled
Workers
Semi-
skilled
Workers
Laborers
and
Servant s
^ ^ (across
Total (vertically
100.00
100.00
26.84
100.00
13.54
100.00
11.83
100.00
15.33
100.00
32.46
100,00
Agriculture, forestt'p and
animal husbandry
26.31
49.2
—
...
...
40.3
Extraction of minerals
2.62
0.3
...
...
8.0
4.1
Mamif act wring and mechanical
industries
30.82
5.9
...
95.4
66.5
23.8
Transportation
7.37
,1.9
8.1 1
4.6
10.1
11.5
Trade
10.20
14.4
56.6
—
5.6
1.6
Public service (not
elsewhere classified)
1.85
i 5.9
...
—
—
0.9
Professional service
6.18
19.1
—
—
0.4
...
Domestic and* personal service
8.16
3.3
...
9.4
18.0
Clerical occupations
7.49
—
55,3
...
...
...
OCCUPATIONS 0? THE GAINFULLY EMPLOYED POPULATION
(10 years of age and over)
.The United States
1920
(Source;- U. S. Bureau of Labor Statistics)
(percentages)
Fig. 463. Here Each Column Totals 100%.
ondary classification, in which the detail figures are percent-
ages of the sub-totals. It is not a matter of importance how
we place these classifications, but as a general rule in tables,
the primary classification should be listed in the column-
headings and the secondary classification in the stubs, to
facilitate checking up on the computing. Where one classifi-
cation is much more lengthy than the other, it is of course
generally more convenient to arrange the longer classification
in the stubs and the shorter one in the column-headings. In
charting, the rule is generally reversed, the primary classifi-
cation being shown along the vertical axis of the chart and
the secondary one being shown horizontally. The chart itself
always shotvs very clearly which classification has been made
of primary importance and which of secondary importance.
It often happens that both classifications appear on their
merits to be equally important, but it is nevertheless necessary
that the distinction be made and the data must be prepared
in the form described before the chart can be made.
The chart is made by laying out a square with co-ordinate
rulings. Along both axes of the square, that is, along its vertical
HUNDRED-PER-CENT S^UHRES
603
and horizontal edges, a scale is marked ofF in percentages from
0% to 100%. Arithmetic projection of the scales is used only
in area charts; and*in the usual form, that is, in the truly
square-shaped chart, both scales are identical. The primary
classification of the grand total is laid off upon the vertical
scale by means of horizontal lines extended across the chart
to form layers of a uniform length but of varying widths or
depths.
At this stage, the chart reminds us of a 100% bar turned on
end and made very short and thick, for the chart bears as yet
MctrmiOMS OP THE OAi>iptn;.i.y employed pOroLAtioa
(10 jtBTt of kg* «nl ov«r)
ih» Dnlt«S States
1920
(Smireei- It. 3. Butveu of Labor Ststlatleili
Fig. 464. The Primary Division Alone Plotted from Fig. 462.
but one classification and its segments or layers show, both
by their depth and their areas, the figures for this one classifi-
cation of the parts of the grand total. The layers, however,
need not be distinguished by colors or shadings, for as will
shortly be seen, they will be sufficiently distinguished by the
markings of the secondary classification. The next step is to
enter these secondary classifications. Each layer is now
treated as a separate 100% bar and divided up as indicated
by the detail figures in the body of the table of data. Notice
that each layer is separately segmented, by vertical dividing
lines which may or may not vary in their positions from layer
6o4 charts and graphs
to layer. These segments of the layers are now colored or
shaded to distinguish then, and the chart is complete. A key
Fig. 465. The Completed Square.
to the shadings should be added, to guide the reader of the
chart. And it will be seen that the area of each shaded seg-
ment of a layer is to the area of the entire square, as the ab-
solute value of each detail figure in the body of the table of
original data is to the absolute value of the grand total.
Various modifications of the 100% square are sometimes
useful. The scale of the primary classification may be cali-
brated in absolute values instead of percentages. In this case,
the square-shaped outline of the entire chart is often discarded,
and the chart made rectangular, thus becoming a 100%
rectangle. This is often done where the detail of secondary
sub-divisions is great and the areas of segments would be so
small as to be ill shown except by enlarging one of the scales.
Either scale may be enlarged in this way, according to the
nature of the data. It is also possible to project the primary
classification upon the horizontal axis and the secondary one
vertically. In this form, the chart strongly resembles the
HUNDRED-PER-CENT SQUARES bo s
staircase relative band curve chart with ordinates at irregular
intervals* The rectilinear lines used to segment the layers
ei«r« mivi Slcuioii Ssifl-
klndroa «fprV*r»
Fig. 466. Here the Primary Division is the Horizontal One,
Plotted from Fig. 463.
form staircase curves separating the colored or shaded bands
which form the sub-totals according to the secondary classi-
fication.
The 100% square or rectangle becomes identical with the
relative band-chart (with staircased curved) when the primary
classification has a numerical basis, forms an ordered math-
ematical series; and can be called a variable. So that the
whole subject of relative band curves charts may indeed be
considered a detail of the 100% area. Frequency series fall
particularly well under either head. And because the pri-
mary classification can be shown upon .either axis of the 100%
area, it often happens that what is really only a relative band
frequency curve chart turned upon edge, seems at first to be
a new kind of chart. When smoothed curves are substituted
for the staircase curves, by connecting the mid-points of the
secondary segmenting lines, of the 100% area, the highly de-
scriptive name of ‘‘marble-cake chart" has sometimes been
used for the resulting picture. This is a very interesting form
6o6
CHARTS AND GRAPHS
of the 100% area. Obviously slight errors creep into the area-
representation in this form, but when the change in the sec-
ondary classification is really gradual and not abrupt, the
chart has gained in interpretative powers. It is commonlv
HUNDRED-PER-CENT S^UJRES
607
useful in the analysis of the component parts of a frequency
series, the plotting of the independent variable (or primary
classification) along th^ y-axis affording greater popular appeal
through the coincidence of increasing numerical values and
rising position upon the chart.
The 100% square can be used for data which is not classi-
fied by, or dependent upon an ordered numerical series and in
which there really are no two interlocking schemes of classifica-
tion, but merely one independent variable. In such cases scales
are useless on the chart and the chart itself is wholly pictorial.
Fig. 468. A 100% Square.
Showing Wartime occupations of the population, U. S., 1918, according to official
estimates; taken from the Annual Report of the Secretary of War for 1919. Total
population, 105,000,000.
It has already been said that area charts must be projected
arithmetically upon both axes, for the reason that only upon
6o8
CHARTS AND GRAPHS
this projection does the area, itself illustrate the product of
the linear dimensions. When a “marble-cake chart,” for
example, is drawn with its primary classification upon a log-
arithmically projected scale, the areas upon the chart lose
their significance, and the chart itself really becomes merely
a chart of frequency curves. The logarithmic projection may
be necessary to straighten the curves or to show parts of the
data in sufficient detail, but great care must be exercised that
the reader of the chart should not, under these circumstances,
attach the slightest importance or significance to areas.
The student who has noted how the 100% bar is particu-
larly adapted to showing the division of a whole into two
UNITED STATES
HTT.
CANADA
IN
EUllOPC
[AUSTRIA
17 .
-SIBERIA
tVt7.
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GiERMANY
cczssi
&R. BRITAIN
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4 <7.
CHINA
iS *7.
UNION OF sSOUTH
AFRICA 1
AU»S T R Al- I A
Fig. 460.
Showing the estimated unmined coal supplies in 1920.
parts (though it can show any number of parts) and how the
100% triangle is particularly adapted to showing the division
HUNDRED-PER-CENT SQUARES
Cog
of a whole into three parts, may now be asking himself for a
chart form which will show conveniently the division of a
whole into four or five parts. In this case he will possibly
have use for a hundred per cent square in which the four or
five segments are indicated only by points and arrows or short
lines. Thus if we are dealing with the sales of the four lines
of a company in many different sales districts, we can combine
them into two groups of two each and plotting each group as
layers, we can indicate the division lines in the layers by short
lines from the layer-division line only for its distance between
WORLD'5 COAL 5UFFLY
(EiTIMATED UNMINEO IN 1920)
CHAN O TOTAL » 000,000 roHi
Fig. 470. Same as the Last in Circular Form.
6io
CHARTS AND GRAPHS
these points. Such a chart reminds us of the famous “swastika”
pattern. The chart is not of much general value, but would so
abbreviate the rulings of the 100% square that many 100%
squares could be superimposed or combined upon one chart
with a visible record for all. Of course, the sizes of the areas
would only have relative significance here, as the total areas
for all grand totals would be the same regardless of their
absolute values.
Closely related to the 100% square is a special type of
100% circle or pie-chart which has recently come into vogue.
By means of an elaborate method of segmentation, small per-
centages and complicated groupings of parts can be shown
without difficulty. For it is obvious that by the simple method
JFWI.SH Population of World
ida.o csTirtAm
&AAa|0 'T»TAik. • IS 000,00 0 '
Fig. 471.
of segmentation of the pie-chart, in which each segment or
part of the circular area- extends from center to circumference,
the small segments or parts become long thin attentuated
HUNDRED-PER-CENT SQUARES 6i i
areas, which are not easily labelled. The more elaborate
method breaks the circular area into concentric rings, the width
of each ring being particularly calculated to fit a particular
segment in the circle, and to result in significant areas within
the segment inside and outside of the ring. The making of a
chart of this kind is not as easy a matter as with the simpler
method, when all segments extend from center to circumfer-
ence. For virtually each angular segment, or slice of the pie-
chart is subjected to crosswise segmentation, and the ring-like
division lines, or arcs, must be placed at distances from the
center which correspond, not to the ratio of the parts to the
whole of the segment, but to the square root of that ratio.
The calculating is not easy. But the chart has very definite
advantages for detailed and minute data when it is desirable
to show several groupings simultaneously. There is little to
recommend it for purposes of precise and comparative study,
but for popular and unscientific purposes it is much favored.
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Fig. 472. A Third Classification has been Added Here by Diagonal
Divisions and Shadings, Showing Sex.
The 100% square or rectangle, and its many variations, are
adaptable to a wide variety of uses. No set rules can be laid
down to limit the various wzys in which it may be applied.
6i2
CHARTS AND GRAPHS
But a very careful study of the results should always be made,
to ascertain that one of the simpler methods in which areas
have no especial significance would not, after all, have pro-
duced more simple and forceful results. The danger is not
that the ingenious chart-maker will fail to utilize all the
possible significant features of the compound area chart, but
that he will utilize too many of them, overcrowding his chart
with complex details. The simpler, the better, both for research
and publicity. And as area charts are more generally popular
in their appeal, simplicity is a cardinal virtue. .
Chapter LI
AREA-BAR-CHARTS
It has been already laid down as a general rule that area
charts (that is charts in which both dimensions of a charted
area vary) are useful only when the data represented by the
area is of less importance than the data of its factors. For
the area chart is based upon the geometrical theorem that the
number of units of measurement in a rectangular area is equal
to the product of the linear units of measurement along its
two sides or dimensions. From this it follows that we can
always show a numerical value by an area whenever we can
break that numerical value into two factors and can plot these
two factors as the two dimensions of the area.
In some cases the presence of two factors in the data or
the fact that the data is the product of two factors, is so
obvious as to be self-apparent. Thus the floor space of a room
is obviously the product of its length and its breadth, and a
chart of the room showing its dimensions and resulting area,
could be constructed by the veriest novice. But should we
come to compare a number of such rooms, it would be a real
question whether to show the dimensions of the rooms or to
show only their total areas, that is, whether to use an area-bar
chart or an ordinary bar-chart. If the figure for total areas
(or square feet) is more important, we must drop these variable-
area charts and present the data of square feet along one
dimension only by a bar-chart. If, on the other hand, it is
the shape or dimensions of the rooms in which we are more
interested, then of course we should adhere to the area diagram
and let the reader rely upon guesswork or upon appended
data for the total area. When both aspects of the data are
important, it would indeed be best of all to use both methods,
outlining the shape of the room by small area diagrams and
showing their comparative sizes by a bar-chart. This example
of data of square measurements of a physical area excellently
613
CHARTS AND GRAPHS
614
illustrates the fact that even data of the most obviously two-
dimensional nature is, so far as the product or resultant is
concerned, best shown by a one dimension chart.
On the other hand, data which seems mosft clearly to be
one-dimensional in its nature, can always, if you desire, be
broken up into two factorial parts. When this is done and
you regard the factorial parts or factors of the data as more
important than the data itself, you can then proceed to show
these factors with their products by an area-bar chart. This
breaking up into factors, it may be remarked, can always be
obtained by a process of division. Thus the sales of our com-
pany in various States may be divided by population of these
States, and so the per capita sales will be obtained. The same
total sales in each State might also be divided by the number
of dealers in each State and so the sales per dealer be obtained.
Or these total sales might be divided by the similar total sales
of the previous year, and so the percentage of increase be
obtained. In short, the most palpably one-dimensional data
may, by the process of division, be turned into factorial two-
dimensional data and shown by the area chart.
Making the chart, it is neither desirable nor commonly
feasible to place the zero line of both axes of all the areas
together, for this would require that they be superimposed
upon each other. The result would be the same as if, in making
the multiple bar-chart, we had superimposed the correspond-
ing bars, for each lower bar would be at least partially hidden
by the upper one and if the upper one were at any time longer
or larger than the lower one, the lower bar would of course be
entirely hidden. This method of superimposition is occasion-
ally used in area charts when the difference between the com-
pared areas along both dimensions is very great. We then
have the effect of squares or rectangles within squares or
rectangles, the inner one being placed at one corner of the
outer one. The reader must then be carefully warned that
the larger area includes the smaller one and is not alone com-
posed of its visible portions outside of the smaller one. In
general, the method is unsatisfactory and to be avoided.
The proper method of showing areas to be compared is to
arrange them side by side, so that along one of the dimensions
only, the area will have a common zero line or base-line.
The result then closely resembles a bar-chart, its only distinc-
tion being that the bars which form the areas are not of a
AREA^BAR^CHARTS
615
constant v/idth, as in the bar-chart, but are of varying widths,
the variations in width showing the second factor in the data.
And it will be seen^hat both the varying widths and the
resulting areas are of secondary importance, serving to give
the reader of the chart a general impression of the relative
importance of the items which are described by the var-
ious lengths of the area-bars. For as has been repeat-
edly pointed out, the reader will have difficulty in precisely
comparing areas of different sizes, and it is obvious that he
will also be unable to gain exact impressions of the various
widths. The most that can be said for the chart is that it
gives him a precise knowledge of statistics of one factor in
the data (as shown by the lengths of the bar areas) — in this the
chart has all the virtues of a bar-chart — and that it also gives
CLOTHING
DIAMONDS
DRUGS
GROCERIES
HARDWARE
UEWELRV \
STATIONERy
SHOES
DRV GOODS
MACHINE TOOLS'
63 .
Figf. 473. A Simple and Excellent Area Bar-chart.
Sales of wholesale concerns in the Second Federal Reserve District in April,
1922, compared with their sales in April, 1921. Width of bars indicates relative
amount of goods sold . — Permission of Mr. Carl Snyder.
him a general impression of the other factor and of the result-
ing product of the two factors — in this the chart is an improve-
ment upon the bar-chart.
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AREA-BAR^CHAR TS 617
The natural arrangement of the area bars would appear
to be in a column down the page. The statistics and labels
or items would then *also be placed in columns to the left of
the bar area, the whole chart closely resembling the bar-chart.
To this chart we may give the name of ‘'area-bar-chart. The
chart is dignified, sound, and extremely illuminating. It
requires but little more labor than the ordinary bar-chart,
for the minor importance of the second factor and the resulting
product, shown by widths and areas of the bars, makes it
unnecessary for these to be plotted with extreme accuracy.
If a scale for the widths be chosen small enough, the data and
labels or items appended to the chart can be entered at fairly
uniform distances down the chart, making for a very present-
able appearance. The small scale upon which the widths of
the bars are plotted deters the reader from attempting pre-
cisel}?' to estimate the secondary or less important factor and
the area, while it nevertheless gives him an excellent idea of
the relative importance of the primary data presented in bar-
chart form. The chart is a direct outgrow^th of the simple
bar-chart, and in its proper place a decided improvement.
By far the more popular form of area-bar-charts however,
is the modification of the pipe-organ or vertical bar-chart.
To this form, the name of “sky-line chart'^ is sometimes given.
In the sky-line chart, however, it is customary to give to the
widths of the bars, a somewhat larger scale, showing their
variations with more precision and emphasizing differences in
areas. Partly on account of the greater widths and partly
for the increased spectacular effect (which is always desired
in popular charts) the areas are placed in direct contact with
each other, being sti'ung out across the page in vivid resem-
blance, let us say, to such a silhouette as the sky-line of lower
New York City as it is first seen by a visitor from abroad.
In this chart distinctive shadings or color tints are often desir-
able to distinguish the areas because of their close contact
with each other. For the same reason very narrow bars which
could only be shown by thin vertical lines are better shown
with narrow separating margins between the lines or in the
form of the previously described area-bar-charts.
In the choice of shadings in these charts, as elsewhere,
where shadings are used, care must be taken to avoid optical
illusions, produced by bringing together shadings of different
color density, for it will be found that two equal areas will
AREA-BAR-CHARTS
619
appear unequal if one is much more densely shaded than the
other. In the sky-line chart care must also be used in the
entering of data and item labels, for the same difficulties of
typography which were met with in the pipe-organ chart may
be encountered here and the same considerations (discussed
in the chapter on pipe-organ charts) in general, apply.
In both area-bar-charts and sky-line charts, it is sometimes
useful to show by a dotted or broken line the contour of the
entire group if all its varying areas were combined into one
area. This light, broken line serves to show the average or
normal or typical phenomenon from which the individual
areas are variations.
When the items or stubs (the independent variable) of the
data form an ordered numerical series, we find ourselves, in
area-bar-charts, back to frequency curves in staircase form.
The frequency curve in staircase form is essentially nothing
more than a sky-line chart, generally of a certain charac-
teristic type (the tallest area-bar being near the center and
those at the side diminishing in height gradually until they
vanish entirely). In the chapter on frequency curves we
have seen that when the items form a continuous and not a
discrete series, a smoothed curve can be plotted through the
mid-points of the top ends of the area-bars making a frequency
polygon. And we have seen that while the total area under
the frequency curve truly represents the total aggregate of
the frequency series, nevertheless the areas between any two
ordinates under the smoothed curve are not individually equal
to the corresponding area-bars in the staircase frequency curve
(unless the adjoining values about any particular item happen
to form an arithmetical series). This inaccuracy, we have
seen, is sometimes more than compensated for by the more
suggestive value of the smoothed curve.
A more complicated form of the area-bar-chart occurs when
the areas are segmented like 100% bars to show a secondary
classification or subdivision. This chart closely corresponds
to the compound bar-chart, and the 100% square, differing
from the former in the varying widths of its bars or layers,
and from the latter in their varying lengths. It is generally
of use in the analysis of the parts of a frequency series, either
cumulated or simple. It is a sort of frequency band curve, in
which the actual values are plotted on both axes, where the
100% square, rectangle, or marble-cake (smoothed vertical
6-20
CHARTS AND GRAPHS
curves) chart projected the .secondary classification only in
percentages. The ''stream chart/’ too, in which the bars or
areas are arranged on both sides of an axis, can be modified
to present similar area features.
In the particular case where the data contains two pairs,
with one product always equal to another product, the two
areas representing these two pairs can be pictorially shown as
suspended and balancing each other upon the two arms of a
chemist’s weighing scales or balance. The two areas must of
course be suspended at equal distances along the lever arms of
the balance. The pictorial representation of the balance
suggests to the reader the equality of the two products.
T. P
'697 CS~
1696 GEI
1699 CZZ
1900 <jEI
1901 (juZ
1902 CEZ
1903 <211
1904 asz
1903 (SZ
(906 (ZZ
1907 CSZ
1906 <ZZ
1909 <SZ
1910 (XZ
191 I <SI
mz <SZ
1910 csi:
1914
19(9
1917
1916
The bl«ck ttJrcAS indicate weights, or counter-poises, the equilibrium of which corresponds to the "equation of exchanie *’
These black areas 'from left to right represent;
M', bank deposits subject to check, m billions of dollars.
M, t e.t money in circulation in the United States (outside of the United ‘States Treasury and the banht), m bilhohs
tif doUan.
T, i e., the volume of trade circulated in billions of "units” (each "unit” 'being that quantity which could he*
purchased for one dollar In 1909).
The lever armt of the above three weights represent:
V', i.e., the Velocity of circulation (“activity") of the deposits,, M'
V, ie., the velocity of circulation of the money, M.
T, i c., the index number, or scale of prices, at which the trade, T, is conducted (This scale of prices is measured
as a percentage of the scale of prices of 1909.)
Fig. 476. Balance or Counter-poise Chart with Two Factors.
The chart illustrates Professor Fisher’s quantity theory of money, according to
which (M'x VO+(M x V) — (T x P) that is, checking deposits times their velocity,
plus money in circulation times its velocity, equals prices times volume of trade.
The weights are here shown as horizontal lines and their factors as leverages. — *
Permission of Mr. Irving Fisher.
AREA-BAR-^CHARTS
621
A more striking method of using the same idea is to show
one of the factors of each product by a weight or bar suspended
from the lever arm and the other by the distance along the
lever arm between the weight and the fulcrum or point of
balance. In this case, we have, as it were, abbreviated the
area, merely showing its two dimensions and leaving the reader
to imagine the area itself by projecting these two linear dimen-
sions. The picture has little analytical value but is a powerful
means of visualizing to the reader the mathematical relation,
AB == X K Professor Irving Fisher has used a series of the
two-factor scales or balances with excellent effect in his exposi-
tion of the quantity theory of money.
This method can indeed be extended to show the equality
of products not of two, but of three, factors as well. Each pair
of ^Talances’’ or weighing ^^scales’’ illustrates an equation of
the form ABC = DEF. And a series of such charts would
illustrate a series of such equations. Since we have used the
radial distance from the fulcrum or point of balance to the
point of suspension to represent one factor, we can show the
weight there suspended as a two-dimension area, the length
Fig. 477. A More Pictorial Form of the Preceding Chart.
A detail of the chart in Professor Fisher’s book, “The Purchasing Power of
Money,” in which the second factors, that is, in the chart, the weights, have
been pictured realistically . — Permission of Mr, Iwing Fisher.
622
CHARTS AND GRAPHS
and width of which illustrate the other two factors. Such
balance pictures need not be the same in pattern on both sides
or areas of the balance; we may for example show the three
factors on one side and their total product upon the other,
using for the latter merely a horizontal bar. Thus total sales
may be shown as a plain bar-chart, each bar being centered at
a fixed point on one side of the balance, and per capita sales
may be shown as an area-bar chart, positioned at the other
end at distances corresponding to the population. The areas
would show a long one dimension the ^^last year’s per capita
sales of the quota’’ and along the other the present percentage
thereof. In short the balance or weighing scales chart is but
a form of compound area-bar chart with especial pictorial
value for its particular relations.
The ingenious chart-maker will be able to apply the prin-
ciples of the area-bar chart in a wide variety of ways, always
remembering that the widths and areas are less significant
than the lengths and should be only used as qualifying or
secondary information serving to evaluate or weigh the im-
portance of the primary information shown by the length of
the bars. In some cases, it is possible to apply these principles
even to such well established bar-charts as the Gantt progress
chart. A step in the direction of this qualifying evaluation
was taken in the chapter on bar-charts when wider bars were
recommended for total-group and sub-total bars in a bar-chart.
There is, indeed, no reason why the area chart principles
cannot be applied to circular graphs. And in the next chapter
the reader will find the same principles extended to wholly
irregular areas, such as map outlines. In general, the area
chart like the 100% square, is essentially popular in its appeal
and simplicity is therefore of first importance. Although every
digression from simple linear measurement results in a loss of
precise legibility, yet when properly used, the principles of
areas can be made to improve a great many charts both in
attractiveness and in instructive value.
Chapter LII
POPULATION MAPS
Every digression from simple linear measurements results
in a loss of precise legibility. In the rectilinear area chart we
have seen that the area itself was but a poor illustration of
the values it represented and was therefore useful chiefly for
the sake of the general impression which it gave of relative
importance of items or values already illustrated by lines.
We are now about to take still another step away from simple
linear dimensions and make use of areas of irregular outline.
It is therefore more than ever necessary to repeat that the
areas have little more than a qualifying or evaluating use,
serving to give a general impre^on of the relative impor-
tance of items.
We do not sell our goods to the mountains, bill them to
the rivers, or credit the forests with payment. Probably from
at least a subconscious appreciation of this circumstance,
many national distributors, advertisers, and sales-managers
have discarded maps on which the rivers, forests or mountains
are shown wheq they are studying the geographic distribution
of their sales. The up-to-date sales manager plots his dis-
tributing points and records his sales in a great many ways
upon maps which carry only faint State outlines or at the most
show the location of the larger cities. But why stop here?
Your sales manager does not sell to square miles, acres, or
other units of land-area measurement. He sells to human
beings. Why should he use maps which show, not human
beings, but square miles, that is, maps in which the areas
indicate not the population but the land surface ? Why indeed !
The average density of the population in the United States
proper at the last census was thirty-five persons per square
mile. This density however varies from State to State. In
some New England States there are more than four hundred
623
CHARTS AND GRAPHS
Fig. 478. Every Map is an Area Chart. On this Map the Areas Represent Square Miles.
Showing that the value of farm-land, shown by the Shadings, is very different in different parts of the country,— From U. S. Census.
POP ULATION i\L 6 1 5
persons per square mile while in some Rocky Mountain States
there is less than one person per square mile.
A handful of peas in the bottom of a box can be kept in a
small corner if you hold them with your hands, but if you
release them, they will quickly spread most evenly over the
bottom of the box almost like water. Imagine the population
which is pent up in these small eastern States, suddenly being
released like the peas in the bottom of the box, and flowing
out over the land of the United States until its density is
uniform throughout, that is eight persons per square mile.
Also imagine the population as carrying its State borders with
it so that the enormous population of the Northeastern States,
spreading out more than half way across the continent, would
carry the borders of the northeastern States westward and
southward with them. (Readers of this book who live west
of the Mississippi river or south of the Mason and Dixon line
may oqiit the remainder of this chapter!)
The result of this projection of the map of the United
States upon a population basis rather than a land-area basis
will be most surprising even to the most hardened travellers.
A comparison of such a population-projection map with a land-
area projection map will show how far the State lines have
been shifted. From a position about one-third of the way
across the map from the Atlantic ocean, the Mississippi river
shifts to a. position about a fifth of the way from the Pacific.
The Rocky Mountain region becomes a narrow strip on the
map. The Southern States shrink frightfully. But if the
familiar outlines of the States are approximately kept in the
new projection, the States will still be easily identified in their
new form and you no longer have difficulty in locating im-
portant but crowded eastern cities.
Needless to say, the picture of sales conditions which such
a map exhibits, will be far more valuable and useful than the
picture upon the usual land-area basis. For in spite -oi a
thorough knowledge of the various State populations, even an
expert on population statistics will find less difficulty in visual-
izing sales conditions as far as the real market, that is the
population itself, is concerned. You will no longer attach
grave importance to the far Western States which show up
poorly on your colored scale map, for they will no longer be
enormous and terrifying red areas. But you will attach far
more importance to the red color when it appears in the
626
CHARTS AND GRAPHS
Fig. 479. The Usual Map of the United States.
628
CHARI S AND GRAPHS
eastern States. Per capita sales statistics, especially, will be
useful on this projection. And the location of sales distribu-
tion points, branch offices, and representfatives or other sales-
men will on this map show how evenly the market is saturated
with your agencies rather than how regularly they are placed
as mile-stones across the country. In short, the corrected
areas of the States serve to give an excellent background or
evaluation of the importance of the statistics plotted upon the
map.
Other conditions beside the total population can be made
as the basis of projection of the map. If your market is best
shown by the native white population, for example, the map
should be projected upon not a total population basis, but
upon the basis of this particular class of the population. If
your market is best shown by the dealers, retailers, jobbers
or brokers, then the map should be based either upon the
number of distributing agencies or upon their aggregate finan-
cial rating. For the analysis of business or economic conditions
relative or comparable to the wheat crop, or any other form of
produce or natural resources, a map might be called for, which
is projected upon the basis of the average yield of this par-
ticular crop or resources during the past years.
The number of ways in which the map can be altered .and
projected for special purposes upon special bases is unlimited,
but all are alike in one respect — that their areas no longer show
physical land areas in square miles but show the actual values
more important for the special purposes in view. This does
not mean that a land area map is useless; on the contrary,
there are many processes, such as’ shipping, railroading, and
travelling, for which the actual land distances are important.
It is only intended to show that where land area is not im-
portant and some other condition is important, it is possible
for the map to represent the values w^e consider important,
whatever they be.
The making of such special projection maps is very difficult
and tedious and it is much better to purchase them from the
few publishers who supply them. "When you must prepare
them yourself, perhaps the best method is to use a large sheet
of cross-ruled paper in which the co-ordinates cut the paper into
small squares of one-tenth of an inch or one millimeter each.
Haying before you a table of the values which you wish to
project as areas on the map, lay out by the rule of “try, try.
POPLJLATlOiS MAPS
629
try again/’ the State outlines, taking care to maintain their
familiar shapes as far as possible, but at the same time counting
the number of small scfuares included in the outline and chang-
ing the outline until the right number of small squares has been
inscribed. For checking purposes, a planimeter (an engineer’s
instrument which measures areas) is useful. If only rough
outlines are required, you might find that a large supply of
differently colored small glass balls could be counted out for
each State, one color for each State, and quickly adjusted into
familiar outlines and closely packed to secure the right area.
Another short cut is to use differently colored plastocine or
children’s modelling wax, and, having weighed the right quan-
tities to suit your data, to mold these into familiar outlines
and press them flat to a uniform thickness. When beads or
wax are used, color distinctions should be maintained for the
different States. Thin strips of paper along State borders
may help to keep the colors from mixing.
The salesmanager will seize upon this map eagerly for sales
analysis. The economist will often find it invaluable. Even
the layman, with no charting or graphic analysis to make,
will find it of absorbing interest. The correction which the
map gives to our conceptions of State populations makes the
map of real educational value, and the school geographies
should have not only national but world maps upon such
projections. The student will note that the principle of the
map is the same as that of other area charts. He will recognize
that precise estimates of the values represented, by the areas
are not possible, particularly as the shapes of the areas are
irregular. He will see that in common with other area charts,
the real value of the areas on the map does not lie in exact
measurements of the values of secondary Importance repre-
sented by the areas, but in the general weighting or evaluation
of the relative importance which is given to the data of primary
importance plotted or recorded upon the map.
Chapter LIII
MODELS
We have now seen used for charts, successively, the point,
the line, and the area. The single straight line, or single system
of straight lines, ending at specified points, forms the bar-chart
in its many forms; if the line is circular, the pie-chart results;
in all of these the points are perhaps the essential feature and
the lines and areas may be considered incidental. A series of
points or dots connected by a line, forms a curve; the plot is
then of several points upon a dual system of straight lines,
called co-ordinates; the outstanding feature here is a line
(called a curve) to which intersect points and inscribed areas
are incidental. A series of such lines or curves may be used to
mark off areas, forming the area chart. But all of these forms
are limited to the use of two dimensions. A third dimension
is not supposed to be present, and is actually negligible, being
no more than the thickness of the layer of ink, crayon, or color
used in making the chart.
We now come to project points, lines and surfaces upward
from the plane surface to get charts in which the third dimen-
sion itself is significant. And as may be imagined, we can
attach importance and primary significance either to the ele-
vated point, the elevated curve, or the elevated surface, to
the vertical lines of elevation or the horizontal lines so elevated,
or to one or another of the edgewise planes supporting the
elevated plane, or to the entire volume itself. In short,
we may use as significant any or all of the various intersecting
lines and surfaces which make up the three-dimensional body,
or the intersect points themselves, or the points, lines, and
surfaces within the body. And there is something new — ^we
can use the cubic content of the three-dimensional body as a
basis of charting. This is a gain in simplicity at the cost of
other things, and with this use of the three-dimensional body,
630
MODELS 631
solid, volumetric chart, or model — call it what you will — ^we
shall begin.
The reader whose mind has leapt ahead of the diverse
forms of area charts to speculate upon the possibilities of three-
dimension charts will not be surprised to find the ‘‘moder’
treated as a type of graph or chart. He will realize further that
the model stands in the same relation to flat-charts as sculp-
ture to pictures. Just as the floor surface of a room may be
shown by the area of a plane representation having length and
width corresponding to the length and width of the room,
so too the cubic content of a room may be shown by the volume
of a solid model having the length and width of the area chart
and the further element of height corresponding to the height
of the room. Just as the area-chart represented by its area,
a product of two factors shown linearly, so too the solid model
represents by its volume a product of three factors shown
linearly. In both cases the scales for linear measures can be
projected only arithmetically to secure the significant repre-
sentation of the resultant or product, shown in square units in
the area-chart; in cubic units in* the solid model.
It is important to note that in the solid model even more
than in the area, the representation of the resultant or product,
though precise enough, is not easily amenable to precise esti-
mation or comparison with other such products, for the human
eye can even less easily judge of the relative values of two
volumes than it can of two areas. Hence volumetric measures
should be used, as square or surface measures, only for data in
which the products themselves are of less importance to us
than the factors which we will show linearly and which go to
make up these products.
The reader who has caught the relation between the curve-
chart and the area-chart in the realm of two-dimension charts
will be prepared for a similar distinction in three-dimension
charts. Comparable to the curved-line chart would be the
curved-surface chart; analogous to the area with square units
of measurement, would be the solid with cubic units of meas-
urement. Such a distinction, however, is not of great import-
ance; in neither two or three-dimension charts is it a hard
and fast division. In the discussion of area charts, we have
already seen that whenever the areas are classified by, ar-
ranged in, or dependent upon an ordered numerical series, the
areas may be fitted together to form a curve or curves. Like-
CHARTS AND CRAPHS
wise in the three-dimensional charts, whenever the solids are
classified by, arranged in, or dependent upon an ordered nu-
merical series, we can fit all the solids together to form a
curved plane. But because we shall not consider as a special
subject the single isolated cubes or solids, we shall have little
use for a distinction between volumetric charts (in which the
unit primarily is cubic measurement); they could be made, but
rarely with profit. The more complicated structure of cubes,
their more difficult presentation and inspection, and the fact
that their outer surfaces hide their inner transverse planes
which are essential parts of them, make such isolated solids
and even sets of isolated solids, of little practical value. Under
this head we shall therefore discard all consideration of seg-
mented cubes. The experimentally minded will be able to
construct not only 100% cubes but even sets of several cubes
and solids of various dimensions, which latter he will be able
to compare by means of either or all of their three linear di-
mensions, their three areas in square measurement, or their
one volume in cubic measurement. For the comparison of
different buildings, engines, machines or other physical equip-
ment or structures, such models, or small replicas may indeed
be useful. But apart from miniature replicas of actual phy-
sical objects (in three dimensions) there would be little use
for such isolated models or sets of isolated models. For math-
ematical statistics, the individual factors are better compared
in separate sets, in bar-charts or curves, and the products are
likewise better charted by linear measures in bar-charts or
curves.
Just as the area, measured in square units, is not significant
in all two-dimension charts, so too, the volume, or cubic con-
tent is not always significant in three-dimension charts. For
the co-ordinates in a curve-chart, for example, need not have
any factorial inter-relation; this is the case of historical curves;
we do not multiply the phenomenon by its date of occurrence
to secure a significant product. Likewise the three systems of
co-ordinates used in projecting a solid, need have no factorial
inter-relation which has a meaning for us; in such cases the
three axes and sets of co-ordinates are merely convenient plot-
ting devices which enable us to distinguish three different
variables in our data, and to study their mutual relations
and behavior. At other times, we may detect a distinctly fac-
torial relation between these variables, and then, of course,
MODELS 633
we can identify the product with the cubic content or volume
of the chart.
The most convenfent division of three-dimension charts,
however, and the one which we shall here follow, is the divi-
sion between three-dimension charts in general, and one par-
ticular kind of three-dimension chart in which the first two
dimensions are used to mark off geographical relations. To
all other models and three-dimension charts we give the name
of frequency surfaces; in these the first two dimensions have
for their scales, any other numerical series, either historical
or frequency. But when the numerical series indicates lati-
tude and longitude upon the earth^s surface, or any geograph-
ical co-ordinates, we call the chart a map and because of great
practical use which is made of maps, we shall consider them
in a separate chapter. As has already been pointed out, the
isolated solid or series of separate solids, which is comparable
to the bar-chart when the latter cannot be converted into a
curve, that is to say, which is not classified by, arranged in,
or made dependent upon an ordered numerical series and so
cannot be joined into a curved surface, will not be discussed at
all. And before proceeding to the consideration of the two
types of three-dimension charts we shall first examine in the
next .chapter, the methods by which the third dimension can
be shown.
Chapter LIV
THE THIRD DIMENSION
There are three ways of preparing stereographs, that is,
three-dimensional charts, which may be called respectively,
the model, the axonometric chart, and the orthographic chart.
The first requires three dimensions physically in space; the
second illustrates three dimensions in a two-dimension plane;
the third shows two dimensions faithfully and seeks to repre-
sent the third dimension by some trick of symbols.
If we elect to use the model, it may be either solid or col-
lapsible. To the solid model, actually built up in three di-
mensions, there is of course little structural difficulty. A solid
From *‘The Construction of Graphical Charts^ hy John B, Peddle^ published hy McGraw-Hill
Book Co, N. y.
Fig. 481. A Plaster of Paris Model.
Model showing the relation between heat units per hour per brake horse-power,
compression pressure, and volume of gas mixture for a gas engine.
634
THE THIRD DIMENSION 635
model can be made of wood, or of many layers of cardboard
or corrugated paper, and can also be moulded of plaster of Paris,
of wax, paper-pulp, <tr other material. Detailed instructions
for the modelling of curved planes will be found in the fol-
lowing chapter. Such solid models are of course cumbersome
and unhandy, difficult to file away, or to carry about from
place to place; and would seem justified only in the case of
extremely important data. Moreover, and this is important,
the making of such solid models requires a great deal of time
and trouble, the equipment not being generally immediately
available for them in the average statistical office.
Special varieties of the solid model will also be described in
the immediately following chapters.! These consist of small
forests of vertical wires, or wooden sticks, placed far enough
apart to allow any individual wire or stick to be inspected.
This is in some respects the model par excellence; it admits
of segmentation, for each wire or stick can be differently
colored through parts of its own length. And most unusual
of all, several models can be combined by placing their wires
or sticks side by side with distinctive coloration. The colora-
tion of wires is usually achieved by stringing colored beads
upon them. The forest model is a more or less laborious
affair to construct, but for sufficiently important facts it is of
ample merit to justify its use.
Collapsible models, as the name suggests, are so made that
they may be folded up or expanded at will. When folded, they
lie flat upon a single sheet of paper and can be easily carried
about or filed away as sheets or folders. When expanded to
occupy three physical dimensions in space, they can be stood
up like blocks or other solids. There are three types of col-
lapsible models, with respect to the mechanics of their opera-
tion. The first type telescopes by means of co-planar hinges,
at right angles to each other, like the folds of a pair of bellows
or the hood of a folding camera. The second type collapses
side-wise upon hinges which are all parallel to each other, like
the partitions in a pasteboard egg-box. The third type never
opens out fully for all parts of the model are hinged together
like the leaves in a book.
^ See also, Brinton, Willard C., Graphic Methods for Presenting Facts, Engineering
Magazine Co., New York. Peddle, John B., Construction of Graphical Charts, McGraw-
Hill Book Co.
CHJRTS JNB GIUPHS
r,j6
In the telescopic model all three dimensions are physically
represented by materials in the structure of the chart and
telescoping is only possible by buckling mp of these materials
across one of the three dimensions. Because of this buckling
of materials the telescopic model is not easily made or oper-
ated, and is generally inferior to the other collapsible models.
In the second type, collapsing side-wise, because one of the
physical dimensions is not represented by any structural
material in the chart, the materials lying on the other two
physical planes can be made to fall together as easily as a
From '^The Construction of Graphical CharlsC hy John B. Peddle, by permission.
Fig, 482, Collapsible Model.
house of cards. If the hinges are on edge the chart folds out
to right or left. In this case it is most convenient to make the
chart of intersecting slitted sheets, the sheets to stand in one
direction having vertical slits or key-ways, halfway up from
the bottoms at the points where the cross-wise sheets intersect,
the latter sheets having corresponding slits down through their
upper halves; and the two sets fitting together, as has been
said, like the partitions of the egg-box in which you buy a
dozen eggs. Obviously all sheets should be of stiff material,
rigid enough not to fall over, strong enough not to tear beyond
the slits.
THE THIRD DIMENSION
637
When this type is made with horizontal hinges, only one
set of vertical sheets is used and these are all hinged, parallel
to each other, upon a single horizontal sheet of pasteboard.
That the various hinged sheets may act as one and maintain
their parallelism at all times, short tie-hinges or keys and key-
ways are used across or near their tops. It is convenient to
mount this chart in a heavy pressboard folder of the standard
vertical filing type, with the back of the chart attached to one
half of the folder and its base to the other so that the chart
is always safely housed. Such a model opens out like the more
elaborate, familiar valentines and works on the same principle.
Similarly the forest model already mentioned can be made
collapsible, paper being used in the place of wires or wooden
sticks. This type is perhaps the best of the precise collapsible
models.
The third type of collapsible model has no structural ma-
terials for two of the three perpendicular planes in the three
dimensions, and, as has been said, never opens out fully. Its
single system of plane surfaces, which should be parallel,
radiates from a common hinge like the leaves of a book. It
is in fact no more than a series of two-dimension charts care-
fully bound together to secure perfect ^'registration,’^ one with
another. But frequently it is sufficient for the study and
analysis of the data; and the fact that it is more easily made,
and handled, and suffers less from wear and tear, strongly
recommends it. Moreover, transparent or semi-transparent
paper can safely be used for this chart, enabling the reader to
note more easily changes in its various parts.
This last form of collapsible chart is also readily susceptible
to commercial publication. In German schools and colleges
such diagrams are sometimes used for the study of parts of
complicated machinery. The student no longer needs to have a
physical model of the machine before him, but can fold back
the diagram of each part to inspect the diagram of the parts
inside it. In medical schools such methods are sometimes used
for the illustration of anatomical studies, the first view show-
ing the outer skin; the second, the underlying nerves; the third,
the muscles; the fourth, the inner organs, with perhaps minor
diagrams or part pages folding back for each of these, to show
the internal structure of these organs; the fifth large sheet
(seen by folding back the fourth sheet which carried these
minor books or sets of small pages upon it) showing the bones;
638
CHARTS AND GRAPHS
the sixth, the rear wall of the muscles, nerves or skin again.
The same method of presentation has been effectively used in
costume studies for the stage (and in children’s toys), to show
upon a top sheet the over-garments and outdoor costume;
upon second sheets, the ordinary or house garments; and upon
third sheets, the undergarments, for various national or
period costumes. There would seem to be no reason why the
same method cannot be used to present mathematical data
in similarly constructed and hinged charts. For these books
or series of superimposed charts, as for the collapsible models,
a strong cover is desirable to protect the parts and the best
cover is generally, as has been said, a vertical filing pressboard
folder.
All of these model-charts have a bulkiness, when solid, with
an added element of flimsiness, when collapsible, that militates
against their general usefulness. Moreover, in their prepara-
tion they are costly of time, and often seem to call for ma-
terial, not readily available in the average office. Except in
the last form, when transparent paper can be used, or in other
forms when transparent celluloid is used, or in the solid model
when glass is used, these charts have the disadvantage that
one part of the chart hides other parts; and diagonal lines,
curves, or planes cannot be readily run through the chart to
give interpolated readings. These disadvantages all disappear
in the next form of three dimensional chart, which we shall
now consider.
The axonometric chart is one in which distances are meas-
ured along three axes which have been represented by lines
within a single plane surface. Every photograph and every
picture of physical objects is such a chart. In paintings,
drawings, and pictures of all kinds, perspective is generally
introduced, to make the more distant objects smaller, that
they may appear to be of the same size. Perspective requires
that really parallel lines be shown as converging toward a
non-existant vanishing point; it makes a fixed scale for any
axis useless and greatly increases the difficulties of drawing.
For statistical charts, therefore, we omit perspective, sacrificing
thereby some of the realistic appearance of our picture, but
giving it constant scales which make charting easy and read-
ing accurate,
THE THIRD DIMENSION
^39
The most commonly-used axonometric chart is the one
with isometric rulings.^ Isometric rulings are those in which
the three axes of the .chart intersect to form equal angles of 60
degrees with each other. Any other angles can be used but
these isometric angles are most convenient and satisfactory,
as they afford the fullest detail along each axis with the sharp-
est possible intersections of all co-ordinates. One of the axes
is vertical for the up and down dimension of the chart; the other
two, at 60"^ to this or 30"^ to the horizontal, represent the two
surface dimensions of the base of the chart.
Isometric drawings are so very useful and so easily made
that they should be adopted whenever possible for three-
dimension charts. The isometric co-ordinates can be left upon
the finished charts to facilitate interpolation and estimates of
the values plotted linearly along them by the reader of the
chart, for the rulings serve to connect the points plotted, with
their scales, in the same way that the Cartesian co-ordinates,
that is, the ordinates and abscissae, connect points upon a
curve with the scales of the curve-chart.
But often the effect of many co-ordinate rulings is so con-
fusing in the finished chart that it is preferable to wipe out all
the co-ordinates save those which have been calibrated with
scales and those along which curves or lines have been plotted.
When the isometric co-ordinates are to be omitted in this way
from the final chart, it is better to rule the co-ordinates in the
first place on blank paper in pencil, as they can then be most
easily removed. The commercially-ruled paper is printed in
ink, generally green or red, and the lines will be reproduced
in photographs unless wiped out with Chinese white. When
the drawings are to be traced, of course it is very easy to omit
the co-ordinates. But isometric co-ordinates are so easily pre-
pared in pencil that for most purposes this is sufficient. .
The scales for the isometric chart axes can be varied, of
course, at will; and the student will naturally seek to adjust
them somewhat to the ranges of the variables plotted. But
when all dimensions, or even the two surface dimensions, are
used to present commensurable values, and the commensur-
able nature of these is manifest, the use of different scale-
moduli or units of distance along the different axes, may
result in an awkward unnatural appearance to the chart
2 See also, Professor Guido Marx in the American Machinist VoL 31, Part 2, p.
701 f and Haskell, Allan C., IIozc to Make and Use Graphic Charts
640
CHARTS AND GRAPHS
unless the angles of the axes are altered and the isometric
principle departed from. It then becomes advisable to select
other angles, which restore the natural appearance by swinging
the whole chart about to one side or the other. The ^first
consideration is, of course, the range of the variables; the
second, the relative detail with which the variations should be
shown. From these the total length of the chart along either
axis can be determined and from the relative detail alone, that
is the size of the unit distance or modulus, the proper angles
of the axes should be determined. When it is desired to vary
the angles for these purposes, the ready-made isometric paper
of course cannot he used and the co-ordinates must all be
especially drawn.
Fig. 483. An Axonometric Chart (Not Isonxetric).
Chart showing relation between journal-bearing temperature, surface velocity,
and heat generated, etc.
It is a distinct limitation of all drawings of solid objects,
including isometric drawings, that they show the object from
THE THIRD DIMENSION
64 T
Ratio of scale-moduli or
length along the three
units of
axes.
Tangents of angles formed
by the right and left-hand
axes with the vertical axis.
Left-hand
Vertical
Right-hand
Left-hand
Right-hand
axis
axis
axis
axis
axis
mz
my
mx
(z)
(x)
1
1
1
tan 60°
tan 60°
(Isome
trie ruling)
2
1
2
8:1
8:7
3
1
3
18:1
18:17
1
4
32:1
32:31
5 1
4
6
5:1
3:1
9 i
5
10
11:1
25:8
From John B. Peddle^ "‘Construction of Craphtcal Charts ”
Fig. 484. Instructions for Axon ©metric Chart Scales.
one view-point only. You cannot turn the drawing over for
diflPerent views of the same object, as you could turn the actual
object itself about in your hand. It is therefore necessary to
arrange the scales of the axonometric drawing carefully to
show the best possible view of the object. High points in the
foreground will hide or obscure lower points behind them. The
scales should therefore be so arranged that the high points will
be set as much as possible in the background, and the fore-
ground be devoted to low points; or that sufficient distance
be allowed behind a peak in the foreground to enable us to
show low points behind it. If, by reversing the direction of
one scale, you can secure this result, the scale should be re-
versed; for the result of reversing a diagonal scale is the same
as giving the object itself a quarter-turn in your hand. The
reversal of the other diagonal scale is the same as giving the
object itself a quarter-turn in the opposite direction; and the
reversal of both axes gives a half-turn to the object itself,
showing its rear face. In general, the best position can be
found by a little experimenting and can, with a little skill,
be determined in advance from an inspection of the data.
It is, however, an advantage of the axonometric chart, not
shared either by the model or the orthograph, that interpola-
tion can be most easily accomplished upon it. For the axon-
ometric chart can be made, if we wish, to show all sides of the
chart at once, merely by using points or dotted lines for the
parts which are apparently hidden from view. And even if
these parts are not indicated, the scales and axes still remain,
from which we can on the finished drawing drop parallels to
any desired point and take a reading. Such interpolation
64a CHARTS AND GRAPHS
can be taken from straight lines in the manner just mentioned
or from rounded contour lines drawn in from various observa-
tions, according to the nature of the problem. The axon-
ometric chart, -and in particular the isometric one, is for most
purposes the most satisfactory method of charting three
dimensions.
A feature which belongs both to models and axonometric
charts is that they may present either staircase or smoothed
surfaces. This is obvious enough from the fact that both are
but series of curves; and curves, as you know, can be in either
form. Moreover, since the curved surface or three-dimension
chart is an interlocking of two such series of curves, one along
and the other across the surface, it is obvious that the same
surface may even be smoothed along one axis and staircased
across the other, as well as smoothed or staircased on both.
These possibilities are not generally open to the charts to
which we are coming, the third type of three-dimensional
chart; in the latter the best that can ordinarily be done toward
smoothing is to indicate contour lines, or lines of equal value,
about each peak and valley — a lateral, if you will, rather than
vertical method of smoothing.
The third method of presenting three-dimension charts
(they can hardly be called stereographs in this case) is the
orthographic chart. In this, as already mentioned, two of
the dimensions are precisely shown, exactly as in a two-dimen-
sion chart, and the third dimension is indicated by symbols.
In its very simplest form, numbers alone represent the third
dimension, the numbers being for this purpose considered as
symbols. But the graphic quality of a number is limited, —
consisting wholly of the number of digits in the number itself,
— and this is not usually sufficiently detailed or legible. We
are therefore prone to seek other symbols to which we can,
attach numerical values and which we can explain to the
reader of the chart in an appended “key” which corresponds
to the scale along an axis. And since no symbols have yet
been found which are capable of the infinitessimal graduations
which a scale affords and are at the same time as easily read
vdth accuracy, we are obliged to restrict ourselves to a few
distinctive symbols. These we use not only for certain set
values, but also for all values nearest thereto, establishing in
this way groups or intervals along the range of variation and
using these symbols for all values within each group. In short,
THE THIRD DIMENSION 643
the use of symbols involves a loss of detail in the presentation
of the third dimension.
Two considerations govern the use of the symbols. The
first is that the groups to which the symbols are attached
should be carefully chosen. This consideration is precisely
the same as applied to the formation of frequency series.^
The groups should if possible contain any round numbers or
bunching"up spots near their centers. The intervals, or
group-limits, should be regular and uniform, if the distribution
appears arithmetical, and as nearly as possible to equal geo-
metric intervals if the distribution appears to be logarithmic.
These are obvious principles which the student will soon dis-
cover for himself. The one thing of consequence is that we
should not, as we may often be tempted to do, divide the series
into groups with equal frequencies. Such a practise is con-
fusing and deceiving to the chart-reader and has but limited
meaning.
The other consideration is that the symbols should be such
as form a natural series in themselves, just as if they were
numbered. This gradation of the scale of symbols should be
such that it is obvious to the chart-reader — the more obvious
we make it, the better is our representation of the missing
third dimension. The reader of the chart should be able to
see at a glance the order in which the symbols fall and there-
after should not need to refer to the key again. Indeed, in
those few portions of the chart where the extreme symbols
are used, it is no bad plan to label the chart itself, right
through the symbol, with the words ‘‘High’’ or "Tow,’’ or with
similar words. When this is done, the chart may be called
self-contained and complete and the reader need only refer
to the key for the numerical equivalents. Needless to say,
however, the key should always be attached to the chart,
showing the symbols and stating the limits of the ranges they
represent.
The position of the symbols upon the chart is of course
dictated by the two independent variables in the data, the
plot of which occupies the two co-ordinate dimensions of the
chart. But since the dependent variable can only be shown
approximately by symbols, not precisely, there is opportunity
for two different methods in marking off the parts of the chart
^ See Chapter XXVII, pages 312-313.
CHARTS AND GRAPHS
644
to be symbolized. In the first place, we may take these parts
precisely as we find them in the data. But such a method leaves
to the chance boundaries of the data the question of what
approximate values shall be shown by symbols for any f>ar-
ticular spot. And as a result the most abrupt transitions of
values may take place between two parts of the chart appear-
ing side by side. Where each part is inherently homogeneous
throughout, the resulting chart, though much confused, is
nevertheless accurate.
But where we have reason to believe that the change from
pne spot on the chart to another is more or less gradual, we
are justified in trying to smooth out the steps between parts
of the chart, so that all intermediate symbols appear between
any two non-successive ones; in other words, so that the change
from one part of the chart to another is as smooth as the few
symbols and the given data will allow us to make it. This
results in a much less confusing picture, and in the circum-
stances prescribed, a more significant one. It is the old dis-
tinction between a staircase and a smoothed curve, with all
the attendant details of loss of absolute accuracy over given
areas and greater significance. Only in the three-dimension
chart of the kind we are considering, the process is called
‘^zoning^’ and the lines which bound the zones are called con-
tour lines.^
This is the orthographic chart proper, familiar enough in
weather topography, where the lines of equal barometric
pressure are called isobars; and the lines of equal temperature,
isotherms. But to apply those principles to the chart, what-
ever it he, is often a difficult task. When data is so scattered
that many contour lines must be interpolated between two
known points, the element of ^^guess’^ becomes large, different
chart-makers will often connect zones differently and it becomes
often a decidedly hazardous proceeding. In such cases the
method need not be employed, or if used, the interpolated
zones may be made disproportionately narrow to limit the
possible error as much as possible.
I Many and various are the kinds of symbols which may be
used. One of these is so close to the mere number itself, which
we have already mentioned, that it may be disposed of first.
This method consists of the use of bars, areas or circles in the
^ Contour-lines may be considered a form of superimposed cross-sections.
THE THIRD DIMENSION
645
place of the numbers. These all have the possibility of infinite
gradation, just as have numbers themselves, and so form an
exception to the considerations just laid down for symbols,
and require no key. They are not wholly satisfactory, however,
for they involve as essential the use of one or both of the
dimensions already given on the chart to other variables.
Hence the symbol cannot be evenly spread over the entire
part of the chart to which it applies and when parts are small
and symbols large, the symbol will cover parts to which it
does not belong when perhaps on the same chart, other parts
are large and have very small symbols, the symbol is likely
to be lost and is not fully graphic.
Permission of Country Gentle man.
Fig. 485. The Wrong Way.
If it is desired to use such symbols as these graphs-within^
graphs, then surely the symbol should not be measured in
areas, such as the squares, triangles, stars, or circles, which
one so often sees used for these purposes. For the comparison
between a large circle and a small one, or a large star and a
small one, cannot be accurately made by the reader. It is
much better to use bars or segments of circles, all requiring
linear measurement only. For in this case the reader can be
trusted to arrive at approximately accurate conclusions, in
spite of the fact that the bars have not been aligned at one
end. The one case in which areas (in square units) should be
646
CHARTS AND GRAPHS
used seems to be the case in which the variations symbolized
seem to increase, not in arithmetical or geometrical series, but
in a series of squares and the special square-root projection
is desired for the scale of the symbolized function or third
dimension.
# *0a, 000 cattle.
<1 150,000 to 200,000 cattle.
O 100.000 to 150,000 cattle.
O M.OOO to 100,000 cattle.
O Leas tbaa 50,000 cattle.
ARK.
•VttuV# V**/
******* W cr-
\j~'s*****<»*jH^^-^
The heavy lloea («*) show geographic dlvislofl*.
From U, S. Ce77ius.
Fig. 486. Somewhat Better.
A logical development of this is the use of many dots or
small circles in the place of one big one for each symbol. By
counting dots the reader can get the exact value of the variable
plotted in each part of the chart, unless the dots are allowed to
become so numerous that they cannot be counted. This last
trick, of putting in a great many dots for a single symbol, is
unfortunately a popular, though pernicious, practise — it is as
if the chart-maker were saying to the chart-reader, *1 have gone
to a great deal of useless labor in putting in all these dots,
now you can waste your time counting them.^'
When the parts of the chart are of uniform size and the dots
are evenly distributed within each part, the dot system is
excellent; for the density of the dots is a graphic guide in itself;
but when the parts to be labelled with symbols are not of even
size, and no significant relation holds between the size of the
parts and their symbols, then unhappily the dot-system falls
down again: for a few dots in a small part of the chart will be
more impressive than many in a large part.
THE THIRD DIMENSION
647
The poly-dot symbol leads us logically to the frank use of
shadings, regardless ^of the number of dots, lines, or other
markings in a shading. And this is ordinarily the most satis-
factory of all symbols. For a few different kinds of shadings
can be easily devised, which are not only mutually distinct,
but also have a definite order of intensity, ranging from solid
white to solid black. These shadings can be laid on with
successive hatchings and cross-hatchings, with a section-liner
or tee-square. They require least work if the shadings are so
chosen that each successive symbol has only an added system
of lines or other markings to distinguish it from the previous
one, for in this case all of the lesser shadings can be put on in
the course of making the extreme shading.
It is an advantage of the cross-hatching symbols that they
do not, as a rule, interfere with lettering which may be also
wanted on the various parts of the chart; the lettering can be
read through all but the solid black or very dark symbols,
and in the latter cases, the symbol can be omitted immediately
about the lettering. It is also an advantage of the hatched
symbols that they can be reproduced, in common with the
methods already described, in a variety of ways, including the
line-cut for printing and the mimeograph for offsetting, and the
blue-print or Van Dyke print for copying.
Closely akin to the hatched symbol, is the solid shade or
tint, the shading proper ranging from pure white through
various grays (made by mixing those two ever-present visitors,
India ink and Chinese white) to solid black. These tints
could, of course, be infinitesimally graduated to suit precisely
the values plotted, but this would require unnecessary work
and could not, through optical illusions, be correctly read by
the chart-reader. It is therefore sufficient to use some five or
six equally different shades which can be easily distinguished
on a key. The method of solid shades is not, however, gener-
ally of enough benefit to warrant its use. It requires some
labor in mixing, the liquid may warp the paper or run, the
symbols are never so distinguishable as hatched patterns, and
the resulting chart cannot be reproduced by line-cut, or any
other method except photo-engraving or photostat.^
The most important and satisfactory symbols possible are
solid colors. These can be made very pale so that lettering
2 The ''Ben Day process” can be used on a line-cut to make it slightly resemble a
half-tone. See Appendix C.
648
CHARI S AND GRAPHS
or even a separate scheme of cross-hatching can show through
them. Transparent inks and water colors, used in color photo-
graphs, can be used. Better than ink or water colors, however,
are wax crayons, the cheaper and waxier the better. They do
not require careful mixing, lay on in even density for each
color and do not wrinkle the paper. After the wax color has
been heavily applied, the chart should be carefully scraped
with a sharp knife fa safety razor blade is excellent for the
purpose) and all the surplus wax removed. A pale tint will
remain on the paper, through which typewriting or other letters
or chart drawings (if previously applied) will show clearly.
The important thing about the scale for colors is that the colors
should be in what is called chromatic sequence, the order of
the colors in that of a rainbow or spectrum. Using five colors,
red, orange, yellow, yellow-green and blue-green will be found
excellent. For more colors a dark red and a blue can be
added. But five clusters or symbols are ordinarily sufficient,
yellow representing “average or normaU; orange, “poor^’; red,
bad’’; yellow-green “fair”; and blue-green, “good.”
Some writers have advised an arrangement of colors by
what they call optical density and have attempted to deter-
mine a color density sequence. These efforts have naturally
and necessarily failed — such a scheme, even if it could possibly
be standardized for different inks, papers, and color combina-
tions, would only result in conglomeration through which the
reader would need the constant assistance of the key. The
only disadvantage to colors is their varying photographic
reproducing powers, blue disappearing wholly and turning
white, while red becomes black. A careful chromatic scale
through the colors from red to blue will photograph as a fairly
uniform scale of grays from white to black. But the chief
advantage of the chromatic arrangement of the colors i$ their
logical significance. The reader of the chart if he be not color-
blind, need only know that blue is good and red is bad and is
at once prepared to interpret all the intermediate colors.
You will see that colors, shadings, and even figures alone,
constitute a dimension in themselves upon the chart. And in
almost all instances where models are made for three-dimension
statistics, the same could be charted upon a two-dimension
chart with the use of colors, tints and cross-hatchings in the
place of the third dimension. That the symbolical presenta-
tion of the third dimension is more a series of approximations,
THE THIRD DIMENSION
649
no precise graduations being possible, has already been ex-
plained. But in most cases these approximations are entirely
sufficient.
^When a more graphic or vivid representation of the same
three-dimensional data is desired, with perhaps more precise
presentation of the exact values of the third dimension, the
isometric or other axonometric drawing must be used. But in
this a part of the data may be hidden behind peaks. If this is
the case and all parts of the data must be visible, then of
course you must fall back upon the three-dimension model
itself, either in collapsible or rigid form. The great time con-
sumed in the making of these and the inconvenience in handling
them makes the three-dimension model, rigid or collapsible,
justified only in the case of very important statistics, but the
ease and convenience of colors and isometric projections make
them of very wide general usefulness.
Chapter LV
FREQUENCY SURFACES
The double frequency series is a type of data which can
invariably be recognized by the form of its tabulation. It is
composed of several columns of figures which have common
stubs, and in which the values of the stubs and the values of
the column headings both form mathematical variables. In
the body of the table, that is, at the intersections of rows and
columns, appear the values of the functions or, in a loose sense,
the dependent variable. The general form is the same as the
table of original data for the 100% square, already discussed,
WKT ABD DRY MOHTHS
bxmimxy of the nunbor cf times each month has been first, second,
third, etc, in order of humidity during the years 1868 to 1906 —
38 years in all. Taken from Xlortr of Croton River, B.Y. at dam.
Jan.
Feb.
Uar.
Ipr*
toy.
Jim©
July
lug*
3ep*
let*
Mor.
Dec*
Wettest
5
8
13
7
1
2
2
Second
1
7
12
6
2
2
1
1
3
S
Third
8
8
4
5
1
1
1
1
2
7
Fourth
8
5
5
8
3
1
1
1
1
7
Fifth
4
4
2
6
9
1
2
2
3
1
4
Sixth
4
Z
1
1 I
7
2
4
1
1
4
6
4
Seventh
4
2
2
6
2
1
2
2
11
7
tlghth
X
1
1 .
2
6
11
1
2
4
6
3
V
Hinth
1
1
11
6
8
1
6
3
1
Tenth
2
1
1
6
8
4
S
7
4
2
illeventh
1
1
4
9
9
9
4
2
Pryest
2
2
2
10
7
11
4
Fig. 487.
but the data is not turned into percentages or products of per-
centages as in that case. It is only necessary that the stub
Cjo
FREQUENCY SURFACES
651
and column heading figures, that is, the two independent
variables, each form ordered numerical series. Whenever this
is the case, the values of the function (that is, the detail
figures of the double frequency series, shown in the body of
the table) can be charted in a third dimension. If you think
of pins stuck into the table upon each figure in the body of
the table, the pins representing the figures by their heights,
you will see at once how this is done.
Let us assume that we are standing in a room of rectangular
or square floor shape. Let us mark off* a pattern upon the floor
of this room, a pattern of criss-cross or co-ordinate lines,
calling those which run the length of the room the A;-abscissae
and those which run across the room the y-abscissae. At the
many intersections of these two sets of abscissae, let us drive
tacks into the floor and into the ceiling overhead and run
strings vertically from the floor to the ceiling. Let us call
these vertical lines the %-ordlnates. At equal distances up
these ordinates, or vertical strings, let us fasten horizontal
strings from ordinate to ordinate above both the ^ and the y
abscissae so as to produce a complete net work of crossed lines
in the room, which would present the appearance of plain co-
ordinate rulings when seen from above or from either side. Let
us assume that in some mysterious way we can wander about
this room freely without becoming entangled in the net work
of strings.
To the ^-axis, or distance down the length of the room,
let us give the values of time, letting the first unit of distance
represent one year, the second the next, the third the following
year and so on, so that along the length of the room we have,
on the ^c-axis, a scale of years. And at each year we notice the
cross-wise lines and the vertical lines are repeated to form a
co-ordinately ruled plane perpendicular to the x-^xis. To the
cross-wise distances of the floor along the y-axis let us give,
for example, a scale calibrated to tens of dollars, and running,
let us say, from zero to two hundred dollars, there being twenty
cross-wise divisions of measurement. To the vertical lines or
the ordinates parallel to the %-axis, let us give a scale calibrated
in hundreds from zero to one thousand, there being ten vertical
units of measurement. In short, to each of the three dimen-
sions or axes of the cubic volume of the room, we can attach
scales of calibrations similar to the scales in ordinary curve-
charts along its two dimensions or axes.
CHARTS AND GRAPHS
^>S:^
Returning to the end of the room to the vertical plane cut-
ting across the room at right angles to the A;-axis and inter-
secting the ^-axis itself at the point of the first year, let us
chart upon this plane a frequency curve showing the nuniber
of sales of various sizes for the concern whose business we are
analyzing for the year indicated on the ^-axis. Then on the
next plane, intersecting the x-axis, the point on its scale for
the next year, let us plot a similar frequency curve for the next
year. And so through all the planes, let us plot frequency
curves, one for each year upon the plane intersecting the point
of that year upon the ;^-axis. Let us plot these curves by
attaching red strings to the network of strings which makes
up the planes.
Having completed the series of curves for all the years,
we can step off and look at the result. The curves are likely
to show great similarity with but slight changes in their exact
positions from year to year. These changes of the positions
of the red-string curves show us the changing nature of the
sizes of sales made by the house. The series of red strings
seems to outline a billowy blanket or irregular curved plane.
If through the series of curves we should attach similar red
strings running lengthwise down the room, connecting corre-
sponding points upon the curves, this blanket or suspended
irregular surface would be more visible. Examining any one
of these new connecting strings, we would find that each one
of them forms the curve of the historical changes in the number
of sales of each size through the various years. In short, the
blanket could have been made by plotting the various historical
curves along the room rather than the frequency curves across
the room. The blanket itself is in fact nothing but a historical
projection, carrying a frequency curve through a number of
years, and yet exhibiting all its points at any point of time.
It should be remarked that the name ‘double frequency
series” given to this type of data is a loose one, used to describe
both truly double-frequency series and historical-frequency
series. The curved plane which we have just plotted obviously
represents a historical-frequency series, that is, the data of a
frequency series carried through a number of periods or points
of time. The double-frequency series proper is similar in all
respects, except that time is not one of the independent
variables. In the double frequency series proper, the data of
a frequency series is carried through a number of changing
FREQUENCY SURFACES
653
conditions or classifications which, like time, form a connected
or variable series. The distinction between the historical
frequency series and the double frequency series is not im“
portant, either in computing or charting.
r "Theory of Siatistta," by G U Vide {fourth edtlion)^ published by J. B. LippincoU,
Fig. 488. Smoothed Frequency Surface.
Showing the coi relation between the height of fathers and sons.
As you will see, the name ‘‘curve’’ is really a misnomer for
this form of chart, for the chart does not merely exhibit a
cui’ved line, but exhibits a series of curved lines forming a
curved plane. The word “surface” is ordinarily used for this
form of chart, though it might also in a loose sense be called a
curve. It is not always true that the succession of curved lines
or true curves can be joined to form smoothed surfaces. It
may be that the phenomenon requires that the joining be made
in staircase fashion. The same consideration applies to the
smoothing of the surface between plotted points as applied in
the smoothing of the ordinary curve. It may even be desirable
to smooth the curves along one axis and leave them in staircase
form across on the other axis. The familiar and most useful
forms of the curved surface are, however, the smoothed sur-
654
CHARTS AND GRAPHS
face (smoothed along both independent variable axes), and the
double-frequency polygon which is in staircase form along both
axes.
WET AND DRY MONTHS Summary of the number of times each monthhas been first, se«sond.
third, etc , in order of humidity during the years 1868 to 1906, 38 years m all. Taken from flow of
Croton River, N. Y., at dam.
Fig. 489. Staircased Frequency Surface.
To make a physical model of these two-dimensional curves
is not difficult, but is rather tedious. Different methods have
been recommended when the solid model is desired. A simple
practice is to cut strips of wire and mount them vertically
upon a board, the lengths of the wire emerging out of the board
representing the s-ordinates, each wire being cut off at the
point where it would intersect the curved plane or the indi-
vidual curve. If the board has been previously drilled with
holes to admit the wires, the holes being at the intersections
of the two sets of abscissae, it is not difficult to erect in a very
short time a forest of these wires, their top ends readily out-
lining the contour and shape of the curved plane. The next
step is to place the board with its wires in an enclosed box and
pour enough plaster of Paris over it to cover all the wires.
The last step is to cut away the plaster of Paris until the ends
of the wires appear, the plaster of Paris being easily scraped off
so as to form a smooth plane. It is also a convenience to out-
FREQUENCY SURFACES
655
line the various horizontal co-ordinates upon the sides of this
solid model and upon the curved plane at the top of the model,
where these horizontal co-ordinates intersect the curved plane.
These co-ordinates can be shown by thin black lines thus
facilitating interpolation and the reading of plotted values
from the model.
Another method by which the same kind of solid model
can be obtained is to plot the individual curve upon pieces of
stiff paper, one complete set of the curves being plotted so
that they can be set up side by side in the same way that the
strings were attached to form vertical planes in the network
in our imaginary room. When the variations of these different
curves are great, it is also of advantage to plot another com-
plete set of curves of the same data upon the other independent
variable, this second set of curves showing the appearance of
the connecting strings finally added in the imaginary room
above discussed. In this case, we have two complete sets of
curves, one for the longitudinal curves and the other for the
latitudinal curves. By cutting slits in the heavy paper as
described in the last chapter, the two sets can be fitted to-
gether as the divisions in the ordinary egg box in which you
buy a dozen eggs.
If the paper upon which the curves are plotted is cut away
at the curve, that is, if you take a pair of scissors and cut
through each plotted curve, the lower halves of the chart will
* have the outlines of the curve for their top edges and when
they are joined together like an egg box, they will form a three-
dimension model which can be collapsed, if a collapsible model
is desired. If a rigid solid model is desired, you can pour
plaster of Paris into this collapsible model, scraping the surface
of the plaster of Paris down to the edges of the paper curves so
as to obtain the smoothed curve plane.
\ When the curved plane is to be in staircase form, it is easily
built up out of blocks of wood. The procedure here is very
simple. You need merely take a long piece of finished lumber
with a square cross section and cut it into strips the same length
as the wires which you would have left standing in making a
plaster of Paris model. The wooden i*ectangular cubes are
then glued together to form a single solid model. There is no
need of projecting the ordinates upon this model, as the edges
of the individual pieces of wood indicate the ordinates, but it
is well to mark in the horizontal rulings around the side of the
656 CHARTS AND GRAPHS
model so that the height of the individual pieces of wood can
be easily estimated from an" examination of the model. Need-
less to say, in all types of solid models the scale calibration for
From R. E. Scoit, in Harvard Engineering Journals by permission.
Fig. 490. A Solid Model — Rounded.
Model showing cost of light in cents per 1000 Candlehours with 40-watt “Mazda”
lamps, for any combination of efficiency and smashing point, where price of
lamp is 50 cents and of current 10 cents per Kilowatt hour.
the horizontal distance should be around the edges of the bases
of the models.
Both the smooth and the staircase curved planes can be
often easily pictured upon isometric or other axonometric
paper as described in the previous chapter, eliminating the
cumbersome and tedious'y constructed solid model. In
general the staircase form of curve-plane is perhaps more
easily projected upon this paper than the smoothed surface.
The paper has the disadvantage of presenting the view of the
FREQUENCY SURFACES
6s7
model from one side only. Therefore care should be used in
the selection of the side from which the model will be seen on
isometric paper, in order to get as much detail as possible,
that is as many parts of the curved plane visible as possible
upon the isometric drawing. The isometric drawing is per-
haps best adapted to symmetrical forms of double frequency
series or to the double ogive or cumulated double-frequency
series, for in the case of ogives the variation can all be seen
from one side anyway. A third dimension can of course be
symbolized upon the ordinary two-dimensional chart by the
use of colors or shading which epitomize the staircase form of
iBt
2nd
f
m
* 4th
fith
€th
7th
eth
4* 9th .
10th
nth
12tb
* «
m »
#
•%v
« •
• • «
• •
0
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• ,
0
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*
t •
• • •
* «
000
000 0
00 0
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0*\
00
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0
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0
0
0 0
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0 0
0,
.V.
0
0 0
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♦ 0
0
0 0
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0 0
0*0
• 0
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0
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0 0
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0 0
0 0*
0 0
0*0
0
•*%
• *
0 0
0
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0 0
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w.
0 0
0 0
0 0
0*0
• •
00 •
0 *
0 0
0 0
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0 0
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0 0 0
00 0
00 0
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00 0*
« *
' • •
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0 00
0 0
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* 0
Jfua Apr Uty Jun Jul 8«p 0«t JteV'
Uonths
lET AKD DRY MOHTHS
SumiBAry of th« nuabor of tl»oi oaob month ha* boon firot, ••eond,
thirds oto.* in ordor of humidity durinc tho yooro 1868 to 1906 •
30 yooro in »11. Xakon fro* flew of Croton’ River, K.Y., at dm*
Fig, 491. An Orthographic Model.
curved plane, or by orthographic lines (similar to contour lines
in topography, or to isothermal lines in weather maps) which
zone off the smoothed curve plane.i
In general the frequency surface has for its two horizontal
dimensions, that is, the axes of its two independent variables,
a rectilinear pattern of co-ordinates. It may be, however,
that upon these co-ordinates a series of irregular shaped out-
^ See Chapter LIV.
658
CHARTS' Am GRAPHS
lines are traced, which are the boundaries of irregularly shaped
areas to which our dependent data (the function or body of
the tabulated double-frequency distribution) attaches. The
map is a special case of this in which the horizontal co-ordinates
mark off longitudes and latitudes and the areas represent
geographical localities. The orthographic chart is a general
case of this, in which the irregular outlines are called contour
lines and the areas are merely zones of equal functional value.
On the other hand, it is not necessary that the horizontal co-
ordinates be rectilinear to begin with; thus we may use tri-
linear co-ordinates, such as are used in the hundred per cent
triangle, for our horizontal plane or base. Indeed when the
100% triangle is used we have a peculiar chart showing four
From John B, Peddle s ''ConHniction of Graphical Charts^ by permission.
Fig. 492. A 100% Triangle Model — Four Variables.
Professor Thurston’s solid tri-axial model showing the efficiency (by height) of
alloys of three metals m various proportions.
variables, three of which combine by addition (or logarith-
mically, by multiplication) to form a constant.^ The solid
built up from a 100% triangle, might by its altitudes (or
s-ordinates) show, for example, the efficiency of foods whose
composition is indicated, as to fats, proteids, and carbo-
hydrates, for example, by position horizontally. Various pro-
^ Cf. Robert Thurston, Glyptic Models, in the Transactions of the American Socle y
of Mechanical Engineers, 1898.
FREQUENCY SURFACES
^59
jections of the horizontal scales (for the independent variables)
are possible, including even a probabilities or double-probabil-
ities projection. These obviously have but limited usefulness,
in more complex cases of equating the phenomena with these
variables, the probabilities projections being designed to con-
vert the ogive surfaces into flat tilted planes, and the other
projections having the same objects for other series, all with a
view to the writing of equations.
It is difficult to adhere strictly to the field of chart making
in these more interesting and important mathematical charts.
We are constantly tempted to wander off into the field of
statistical methods with which these charts are sometimes
intimately connected. And though this book is not a manual
of statistical methods we shall here digress long enough to
sketch in a few of the more important uses of the double
frequency curve. For in the statistical laboratory the fre-
quency surface is often used for the study or presentation of
correlation and association between different methods of classi-
fication of the same phenomena. Correlation between two
historical series can, as you have seen, be best shown by the
juxtaposition of their rate-of-change (i.e. logarithmic) curves.
But correlation between two independent variables of the
same data can be shown in detail by the stereographic or
three-dimensional model.
The nature of the normal curve of error, that is, the dis-
persion about a central or most typical point, which is to be
expected under the operation of the law of chance variations,
has already been explained in so far as variation along one
dimension is concerned. But variation can equally well take
place along both dimensions when the nature of the phen-
omenon is such as to permit it. Thus the dispersion of gunfire
from cannon varies as to both distance and direction (range
and deflection). The coaction of the same probable dispersion
along both axes, that is both longitudinally and latitudinally,
results in a cone-like peak whose sides appear to slope along
the curve of normal error, when seen from any side.
When the double-frequency curved plane presents this form,
the thought is naturally suggested that the two independent
variables upon which the grouping or classification of the
functions depends, are really independent in their action, and
do not affect each other. When, instead of a cone with its sides
following the normal curve of error, we have a ridge diagonally
66o
CHARTS .AND GRAPHS
across the chart, whose cross section may or may not resemble
the curve of error, we have a rough means of measuring the
correlation between the two bases, the very narrow ridge pre-
senting high correlation and the wide-spread irregular ridge
showing low correlation.
From G. U. Yule, “7'heory of SiatishcsF fourth edition, published by J. B. Lippincoit.
Fig- 493. The Normal Frequency Surface — Rounded.
The two-dimensional curve or curved plane, either -
smoothed or stepping, is perhaps less often used in business
than it ought to be. It gives to important data a valuable
projection, changing through time and conditions in the case
of historical frequency series, and illustrates the co-action of
the two independent classifications or changing conditions in
the case of the double-frequency series. The labor of prepar-
ing the charts is not great and the illuminating pictures they *
present are ample recompense-
Chapter LVI
RELIEF MAPS.
It is in the more elaborate form of maps that we find the
most frequent and perhaps the most generally understood
form of three-dimensional charts. Every one is familiar with
the relief-map rriodel used in the school room, in which the two
horizontal dimensions are used for the latitude and longitude
as in the ordinary map, but in which the varying heights or
altitudes of the model indicate the altitude of the land,
mountains being shown by ridges, and rivers and valleys by
cuts and hollows.
Business men and economists have perhaps little interest
in the physical contour of a country, but the principles of the
relief-map can be used to illustrate a large variety of other
things than the actual height of the land levels. The sales
manager may be interested in a relief-map in which the %-
ordinates, that is, the vertical distances or heights of the relief-
map model indicate the density of sales, as shown by per capita
sales, per dealer sales, or by other means. The engineer may
be interested in a relief-map in which the height indicates the
amount of natural resources, water power, mineral deposits,
and so on in the various localities. The economist may be
interested in a relief-map showing the financial resources,
wealth, crop yields, or other sociological conditions in the
locality.
The usual way of presenting these relief-maps upon flat
surfaces is to indicate the height which the actual relief model
would have in its various portions, by different kinds of
colors or shading. If colors are used, they should be arranged
along a color chromatic scale so that the colors themselves,
by their changes (for example, from red to blue, through
orange, yellow, yellow-green and blue-green), have a natural
significance and can easily be understood. If the colors are
carefully chosen for their tints and intensities, they can be
66i
662
CHARTS AND GRAPHS
successfully photographed, 'and will show on the photograph
as black for the reds and white for the blues, and varying
through dark grays to light grays for the intermediate colors.
The problem here is to secure color tints which have not
increasing optical intensity but increasing actinic intensity,
for the camera does not photograph different colors precisely
as the eye registers them. Many attempts have been made to
adopt scales of increasing intensity of color, regardless of their
chromatic sequence, but these attempts are almost always un-
successful because of the extreme variation of available colors
which may be used with apparently the same optical results,
but with actually different actinic or photographic values.
Moreover, the arrangement of colors solely according to ocular
density or intensity is unsatisfactory because for the signific-
ance of each color the reader of the chart must refer to a key.
There are many different ways of applying color to charts.
The disadvantage of water colors is that they tend to run
upon the paper and are difficult to shade off, no two mixtures
being precisely alike when laid at different times on different
charts. Colored water-proof inks, ordinarily used in drafting,
must often be diluted or their intensity will be so great as to
hide any printing or labelling which was intended to show
through the color on the chart. For extreme transparency,
photographers' Japanese transparent inks and lantern-slide
colors can be used. Perhaps the best results are obtained in
general with ordinary wax crayons, the cheaper and more waxy
they are, the better. These can be laid on very thickly and
evenly, and can then be scraped away with a sharp knife
edge, leaving a delicate tint through which all printing and
labelling will be easily seen, and not warping or wrinkling the
paper.
Shadings are of many different kinds. The Census Bureau
makes great use of dot shading, a number of small dots being
placed upon the paper, scattered over the locality and by their
number, showing the values of the data for the locality. The
method has a decided advantage in that the charts tend by
their crowding or scattering to indicate density visually, but
it is a tedious method to follow in the making of the chart and
the results do not afford any degree of accurate reading. It is
impossible for the average reader to count the number of dots
where these have been thickly placed and where they are so
thick as to form almost black areas the significance of the
RELIEF MAPS
663
dot has become entirely pictorial. The method is, however,
far better than another one which resorts to dots or circles of
various sizes in which.the areas inscribed in the circles indicate
the yalues of the data. For in these circles of various sizes,
we meet with optical illusions and diiSiculties of accurate chart-
reading as well as chart making, described very early in this
book. The area of the circle although a two-dimensional
measure, is used to illustrate one-dimensionaf data. Dot-
maps are often most easily made with colored map pins or
map tacks which have been described in an earlier chapter.
Much the better form of shading is secured by a careful
scheme of hatching and cross-hatching lines so arranged as to
give an optical effect of shading from white to black and at
the same time sufficiently different in pattern to be easily
identified from a key or appended scale of shades.
Several such patterns of useful shadings have been designed.
The work of drawing in these shadings, however, is sometimes
very great, it is difficult to rule these hatching lines uniformly
without a special instrument (section-ruler) and the whole
process takes a great deal of time. It is therefore sometimes
better for the average chart which must be prepared in black
and white, to mix India ink and Chinese white in various
degrees and ink them onto the chart as so many tints of gray.
When this is reproduced photograjlhically the effects are en-
tirely satisfactory for photostats, but are useless for blue-
prints. As these tints require half-tone engravings for printed
reproduction, it is more convenient, when the map is intended
solely for printed reproduction, to use the forms of cross-
hatching and shadings knov^n as ‘^Ben Day’’ in the printing
office. When Ben Day is used, your original drawing need
have no shading at all, the various types of Ben Day merely
being indicated by numbers or symbols in blue pencil on your
drawing; the engraver will insert them properly.
Relief-map models may be compared to the smoothed-
plane curve of the last chapter, in that the changes of types
are never abrupt but ai'e gradual. The staircase form of these
map models is an elevation map or table-land map which is
less often seen. It can be best constructed with a few sets of
maps, mounted upon boards of different thicknesses, and cut
or sawed apart along the State boundaries. Children’s puzzle
toys are sometimes made in the form of maps of the United
States in which the individual States have been cut away in
ClIJRTS Jj\D GRJPHS
66 ^
this fashion, and may prove useful for this purpose. A picture
of the elevation or table-land map is, however, easily drawn
upon isometric paper, or upon plain paper, by tracing the
State outlines from a regular map of the United States, after
shifting the position of the tracing paper slightly to correspond
to the representation of height or elevation of each State.
Such maps are very effective in their way.
One disadvantage of the colored or cross-hatch map upon
paper drawings is that it represents a staircase form of map
while most phenomena should really be smooth, as the transi-
tions from State to State are not abrupt but gradual. To meet
this problem, the colored or shaded map can often be skilfully
converted into a zone map, in which the colors or shadings
have been zoned so that no two colors or shadings appear side
by side upon the map except those which are consecutive in
the key or scale of colors or shading. Where the data applies
to the entire State or other territory, these zonings are of course
arbitrary and tend to alter significant areas in precisely the
same way as the smoothing of a frequency polygon destroys
significant partial areas of the polygon. The zoning should
therefore be made very narrow, along the outlines of the
States, in order to leave as large as possible a portion of the
space properly colored according to the data,
i ^ However, if the data does not represent the entire State
or other territory, but merely indicates conditions at certain
points, such as certain cities, the zoning should always be done”
and the zones should be of equal width between any two
observed points. A familiar example of this type of zone map
is to be found in the map used by the Weather Bureau showing
high and low pressure areas from day to day and temperature
lines across^ the country. In fact the entire zoning process is
merely an attempt to reproduce for the particular data, the
same excellent results achieved by these isothermal lines upon
the weather map. The contour lines upon topographical maps
are examples of the same type of zoning or orthographic rulings.
The technique of these various presentations of the third
dimension upon the map is fully described in a previous chapter
and we have here only hastily recapitulated the more common
forms of maps. In the case of all the maps so far considered,
the reader will notice a limitation, in that each map is capable
of presenting but one set of data. Two figures for each locality
cannot be shown upon the same map. There are, however,
RELIEF MAPS
665
two ways m which more than one geographical distribution
can be shown upon the same map. The first of these methods
IS the very obvious-one of combining on the same map two
methods of showing the third dimension. Thus on a flat map
both colors and cross-hatchings can be used simultaneously,
the colors to show one set of variables; the hatchings, another.
The results are not wholly satisfactory. If bars or other area
charts (circles, stars, etc.) are used to show values of course a
series of bars or areas can be used in each part of the map,
corresponding bars being perhaps distinguished by color or
shading. If a stereographic map (i.e. solid model or axono-
metric drawing) be used, of course one set of values is shown
by the altitudes (ss-axis ordinates) and’ another by colors or
shadings drawn upon the resulting surface. And always the
use of numbers actually entered upon the map may give us
further values not graphically displayed. All of these methods
give us what might be called multiple maps, in that they are
combinations of two or more map surfaces. The second way
of showing more than one geographical distribution is more
laborious in construction, but also much more illuminating.
It is the method of the bead-map. It gives us not only multiple
maps, that is combinations of distinct distributions, but also
it gives us compound maps, that is segmentations of a single
distribution. This last feature is one which cannot be satis-
factorily achieved by any other graphic methods and can be
showm only on a flat map by figures. The bead-map must be
classed with solid models, for it is a rigid body occupying space
in three physical dimensions as fully as if it were of plaster-of-
paris, wood, or some other substance. In its construction we
fall back upon the upright wires which were used in the con-
struction of the plaster-of-paris model.
The bead-map takes its name from the fact that after
properly cutting the vertical wires, it is customary and most
satisfactory to string beads upon them before their ends are
inserted in the map. Beads can be obtained for use in this
way, in many different colors, at the average department
store. Sometimes glass beads of uniform sizes, especially
advantageous for this work, can be obtained from the publish-
ers of charting material. The wires themselves should be fine,
of medium strength and spring and can also be obtained from
charting-material publishers, especially adapted for this work.
Where only a few beads are to be strung and the wires do not
666
CHARTS AND GRAPHS
extend very far above the paper, very long and thin steel pins
such as are used in natural history museums, can be used for
the wires, the heads of the pins holding the beads and prevent-
ing their escape. When wires are used, small knots must be
tied at their upper ends, to prevent the beads from coming oifF
the wires. The beads on each wire should be all of one color
except that every tenth bead, or (according to the scale for
beads), every 'significant bead, should be of a contrasting color
so as to facilitate the counting of them by the reader of the
map.
These maps can be made with as many as three or four
sets of separate wires for each State in the Union, each set
representing a certain figure or set of data, on maps which
^hemselves are no larger than ordinary-sized letter paper, that
^s, SJ by 11 inches. The different wires for each State should
be placed upon a single line or row across the State so that they
can be easily compared and form, as it were, a vertical bar-
chart upon the State.
The variety of purposes to which the bead map can be put
is as great as the uses of other relief or color maps. The sales
manager, for example, will be interested in a map in which
there are four columns of beads in each State, a column of
green beads indicating the population or potential market in
each State, a column of red beads indicating the sales of his
competitors in the State, a column of blue beads showing his
own sales in the State in the previous year, and a column of
black beads showing his sales this year. A comparison of the
red and black beads shows him how well he is keeping up with
his competitors, while a comparison of the green and black
beads shows him how well the market is being saturated by
his goods, and a comparison of the blue and black beads
shows him his last annual increase or decrease of sales.
Another convenient form of this map is sometimes called
the ‘Tree-map/' In this the stringing of beads upon wires
has been eliminated entirely and small pieces of colored wood
sticks substituted for them. A full equipment of wood sticks
of uniform thickness and shape, but of different lengths and
colors, can be obtained from the manufacturers of kinder-
garten toys, being often sold for kindergarten woi'k. When
these sticks are being used, their ends, to be inserted in the
map, should be sharpened with a knife so that they can be
easily inserted, after first marking upon each clearly the dis-
RELIEF- MAPS
667
tances which should be left exposed, sticking up out of the map.
After they have been driven into the map to the proper dist-
ance, they should be removed and into the holes made by
them drops of glue should be placed, the sticks being then
replaced in the holes and allowed to set in the glue. The tree-
map does not afford the possibility of exact reading which
was possible in the bead map, where the beads themselves on
any wire could be readily counted. If this feature is desired,
small colored bands should be drawn at the points about the
sticks of wood at the heights where the distinctly colored
beads would appear, marking off on each stick of wood the
ordinates of the various convenient values on the s-axis.
Needless to say, the map for beads and wires, or for sticks
of wood, should be mounted in the same way as pin maps,
which have been described in an earlier chapter. They should
have at least three layers of corrugated pasteboard under them
to hold and protect the ends of wires or sticks.^ Neither
bead-maps nor tree-maps are convenient to file or to have
about in large numbers as they are apt to get damaged. The
best way to file them or to carry them about is to use small
wooden boxes or cases into which they can be slipped easily
and fit compactly, and in which they are prevented from
moving about by small retaining flanges inside the boxes or
cas'es.
^ See Chapter IV.
668
CHARTS AND GRAPHS
Keys should always be -provided with every map, to
explain the significance of colors, shadings or beads, and wooden
sticks. These keys serve the same purpose as scales in curve-
or bar-charts. They should be complete and carefully worded.
It goes without saying that the special projection maps de-
scribed in the chapter on population maps, can be used in the
place of the ordinary land-area map for all cases where the
significance of map areas is better shown by such projections.
Maps need not be used only for the display of character-
istics of entire localities and territories, but can also be used
for the analysis of routings and conditions along certain
routes or at certain points on the map. In this case we are
not concerned with areas on the map but with lines or points
upon them. The use of strings upon maps, connecting map
pins or map tacks, has already been described in an early
chapter. These strings can be used not only to indicate
actual routes which will be followed by sales managers,
travellers, or traffic, but can also be used to indicate spheres
of influence, authority, or other connecting influences. Thus
the circulation of a number of newspapers situated at different
points of the country can be shown by strings (or indeed by
mere ink lines) radiating from their places of publication to
the residences of their furthest subscribers, different colored
strings (or ink) being used for each newspaper. Such a map is
sometimes useful in the analysis of newspaper circulation for
advertising campaigns. Likewise the line of authority from
central office to the various branch houses and from branch
offices to the individual agencies can be similarly shown by
radiating strings. Such maps properly belong to the class of
combinations or superimpositions of route-charts upon maps
described in the chapter on combinations of non-mathematical
charts.
When, however, we attempt to show the volume of traffic
or travel, or the extent of any other connecting phenomena
between two points upon the map, we come quickly into the
field of three-dimension maps. If we wish to present this
graphically, a very effective method has been found of pre-
paring ribbons of stiff, colored cardboard or paper, and mount-
ing these ribbons on edge along the route or line of traffic or
connection, so that the height to which the ribbons rise, will
indicate the volume of traffic or other connecting phenomena.
The same result can be more easily obtained by the use of
RELIEF APS 669
colored strings connecting columns of beads which have been
previously erected at the points to be connected. The strings
are easily tied to the columns of beads and run back and forth,
one string between each layer of beads, forming fences similar
to the old-fashioned rail fences and indicating by the number
of strings or rails the volume of traffic or other figures for the
connecting phenomena. If we wish to present the three™
670
CHARTS AND GRAPHS
dimensional data upon a flat surface, for convenience in
handling and filing, and the number of routes or connection
lines is not great, we can show the comparative height of the
various ribbons or set of strings by colored or shaded bands
drawn upon the map connecting the points which the bead or
string fences would connect. In this case, the widths of the
bands representing the volume of traffic or other connecting
phenomena.
The reader will have seen by this time that map-charts
are almost a field of charting in themselves, with wide diversity
and flexibility and an infinite variety of forms, capable of the
widest variation and adaptation for special purposes. In fact,
the map can be considered as distinct from all other types of
charts in that the fundamental two dimensions, that is, the
two dimensions of the base-map itself, are used solely for the
purpose of displaying geographic position and location (except
in population maps) and not for strictly mathematical relations
and that the mathematical relations must be charted upon
this ground-map by the use either of a third dimension or of
superimposed drawings representing a third dimension. In
a sense therefore, the map can be considered an inefficient or
wasteful type of chart. For economy of space and charting
dimensions, the superior form of chart for all data having a
geographical basis is the bar-chart in which the geographical
location is shown by a list of stubs and the independent
variable, that is, the geographical location, occupies only one
dimension on the paper. This applies not only to maps, but
also to diagrams, floor plans and other illustrations of space
or physical localities, all of which can be treated in the same
way as maps have been treated in this chapter. But in spite
of these disadvantages and inconveniences, the map is so useful
that it can be strongly recommended to the studious chart-
maker as a powerful method of displaying such facts as are of
sufficient importance to justify the greater labor and care
involved.
PART VI r. CONCLUSION
Chapter LVII
THE STATISTICAL MATERIALS
Few, perhaps, of our readers, have run the gamut of chart
forms and methods which we have described in this book
without realizing that there is almost as surely a natural
evolution in charts as there is in other sciences or arts. It is
possible and would indeed be interesting to construct a dia-
gram in the form of a tree-chart, showing the development of
each chart iorm out of common root-forms. Within the
bounds of our limited ability we have constructed this book
after such a pattern. And as time passes and new forms are
invented, or new modifications are introduced, it will always,
doubtless, be possible to relate them to existing forms and allo-
cate each to its proper niche upon such a diagram.
But more interesting than the classification of chart-
forms is the classification of the statistical materials which
they illustrate. For the numerical arrays and tabulations,
the counts, samplings, enumerations, and reports which we
call statistics present even greater variety and heterogeneity
than the charts by which we may picture them. Nor need
such a classification be wholly academic. The coding and
systematizing of graphic methods can hardly progress far with-
out becoming tangled in the chaos of statistical forms, a con-
fusion from which it cannot again escape until we have set to
order the statistical stock-room.
If then, we could succeed in so neatly classifying and
pigeon-holing each type, species and hybrid form of statistics,
that the novice could readily identify each specimen, we would
set for ourselves this aim: That each variety of ‘^statistic’’
should be clearly labelled and marked with the one, two, or
more ways of charting suitable for its illustration. We would
have this code, key, or system, so simply set forth that the
economist and the business man, be he ever so untutored in
the science, could easily locate in it his particular bit of statis-
671
67 '2
CHARTS AND GRAPHS
tics and as quickly set out elFectively to chart it. This we say
would be our ambition/ were it possible. And while many
may doubt its possibility^ yet in this chapter we shall venture
a few first steps in its direction, only bespeaking in our readers
a tempering of judgment with generosity for the short-comings
and failures which attend us.
We do not progress far in statistics without noticing that
all numbers are purely adjectival, and that to each number,
in order that it may have a meaning, a substantive must be
attached. If we only make mention of so small a thing as
two pins we may observe that the numerical adjective ^^two’'
holds meaning only when attached to the noun ‘‘pins.""^ Speak
of three needles and we note that in this beginning of a statis-
tical collection we have changed both adjective and noun.
To make it look quite professional and uninteresting we should
tabulate it, thus:
Pins 2
Needles 3
But suppose that these pins were ordinary household pins,
while we may discover elsewhere five safety-pins, to be added
to our collection of pins, bringing it up to seven, thus :
Pins, common 2
Pins, safety 5
Needles 3
Now we notice that while the noun has remained the same,
another adjective has been added beside the numbers. Many
such qualifying additions could be made. And it is the object
of this homely illustration to point out that while numbers
are always adjectival, not all adjectives are numerical, and
that numbers like other adjectives, such as ‘‘safety’^ or
‘‘common, have a meaning only when associated with some
substantive.
But we stand too long on pins and needles. Let us only
keep in mind the point that in all statistical work we must
early recognize two variables, variates, or variable facts,
which for convenience we have always distinguished as inde-
pendent and dependent. It may be that at times one will
seem the noun and the other the adjective, or that various
adjectives will vie with each other for importance, and that
at other times in the same data a reverse arrangement will be
more useful to us, but always at one particular time we must
use one variable as independent, that the other, clinging to it,
THE STATISTICAL MATERIALS
673
may oe dependent. And statistical technique is largely a
matter of the proper marshalling, commandeering and buffet-
ing about of these two^sets of variables until they behave in a
way that yields up to us an intelligible message. There is no
hard and fast distinction between the two variables and their
degrees of independence or dependence — they are truly inter-
dependent — ^but always there are hard and fast rules which
govern the treatment of them in such a way, that whichever
variable is playing the independent role may be given such
and such handling, and the other which is playing the depend-
ent role has such and such other possible operations. As a
rule we usually let the independent variable take its own
course and put most of our efforts upon the dependent one,
but this is not always so.
Another thing which we may note at once is that statistics
may come to us either in singular or plural form. We may
have, as it were, a single ‘"‘statistic,^’ or a collection of statistics.
Thus the simple fact:
Pins 7
may be the alpha and omega of our desired information; or the
more elaborate statement:
Pins, common 2
Pins, safety 5
may' be our objective. Here we come upon the distinction
between what may in a wider sense be termed ^‘averages” and
^distributions.’’ Commonly, this particular average would be
called a ‘‘total,” for it is the total of its parts, which parts are
forcibly brought to mind by the subsequent distribution. And
in a specialized sense, averages and totals are very distinct.
Thus we would say:
Pins, total 7
Pins, common 2
Pins, safety 5
Pins, average 3}-^
But in a wider, perhaps more precise sense, all so-called totals
are merely averages — if you will, averages of the counts taken
of the items, or averages for particular selections or samples.
Into the various kinds of averages we shall not delve — their
consideration forms the large part of most elementary treatises
on the science of statistical methods. Let us compromise by
distinguishing at once single and collective statistics as “aver-
ages or totals” and “distributions.”
674
CHARTS AND GRAPHS
Now before going on to collective statistics, or distribu-
tions, let us look just a little longer at the single entry or item:
Pins . 7
We note that the number, 7, is a count. But a count of what?
A count of the number, you say, of whole pins. So it is. And
being so, it is so obvious that it has not been included in the
statement. Could we count these pins in any other way?
Suppose we xrount the half-pins, then there are fourteen in all.
But this is quibbling, you say. Very well, suppose I inform
you that these pins are exceptionally rare and are valued at
a dollar apiece. Now you can count them as:
Pins ...37.00
or more fully:
Pins (value) 7 (dollars)
Again we discover that they are railroad coupling pins and
weigh two pounds each. Now we can write;
Pins (weight) 14 (pounds)
And all the time we are merely describing in different ways
the same objects! In short, in statistical work, we deal with
various numbers and by them we count various items in terms
of or by means of various units of measurement. The units
may be, roughly, measures of volume or of value. Volume
statistics refer to physical volume in a loose sense and may be
in terms of linear, surface, content, weight, or other measures,
including the mere count of the number of items. Value
statistics refer to intrinsic worth as indicated usually by actual
or potential price or cost, and are generally in terms of money
of one currency or another.
While the separate or isolated average or total may appear
in any of these forms, it does not in any of them become a
proper subject for a chart, for a chart is only useful as a means
of comparing two or more figures. We do not enter the field
of charting therefore, until we come to collective statistics or
distributions. In turning to these, however, we must keep in
mind the various possible forms of the individual item or
^^statistic’^ for it is obvious that they apply as well to the collec-
tion. The collection or table of items, then, will contain both
independent and dependent variables, and will be composed
of either averages or totals (now become sub-totals), measured
in units of volume or value. In fact, the collective statistical
statement often appears to be no more than a mere agglomera-
tion of individual statements placed together upon a page.
:n-IE n'ATlSTlCAL MATERIALS 675
At other times the collective statement appears to be a more
detailed elaboration of some individual statement. In the
latter case the name distribution is clearly called for, but it is
also true that the most patently heterogeneous aggregation or
conglomeration of figures can generally be regarded as a de-
tailed distribution of something or other. In this sense we may
speak of the simple individual item as “undistributed” and the
collection of items as “distributed.”
It is in studying the various types of distributions that we
strike the first important distinctions of data, and, by corollary,
of charts. Four types at once come to mind, which, for con-
venience, we may call respectively, abstract, geographical,
frequency and historical distributions. Indeed it is not im-
probable that in the course of time the compilers of statistical
volumes will adopt distinct tabular forms for each of these
types and consistently maintain these forms in their compila-
tions. The Bureau of the Census has already adopted a more
or less standard form of table for geographical distributions
where these cover the States of the United States. Even
more successful is the excellent standardization of historical
tables by the same bureau for its “Survey of Current Business.”
By the side of the usual confusion of tables in most statistical
compilations the simplicity and clarity of these forms is indeed
refreshing.
There is very little overlapping of the four types of distribu-
tions. The first two are essentially logical, the last two are
essentially numerical, in respect to the bases of their distribu-
tion or classification, which form the Independent variables or
“stubs,” in them. The first and third are generalized types, the
second and fourth are merely extremely common and important
species of the first and third. The basis of the second, the
geographic distribution, is space, while the first, the abstract
distribution, can have as a basis, any other set of logical rela-
tions. The basis of the fourth or historical series is time,
while the third or frequency series can have as a basis any
other set of numerical relations. And, of course, we can have
what might be called composite statistical tables presenting
two or more of these distributions simultaneously. Indeed, in
most statistical compilations the composite distribution is
chiefly used for its convenience of comparisons and economy
of space. Not only can two abstract distributions be made to
interlock in a single composite one, but two geographic and
676
CHJRTS JND GRAPHS
two frequency distributions are often so combined. And
there can be any combination of different types. In the present
book the distribution placed at the sides^ of the table has been
called the series of stubs, generally considered the more im-
portant, and the other, placed across the top of the table,
has been called the series of column-headings or captions.
If we should attempt the explicit description of statistical
distributions* of these various types — confining ourselves to
single distributions only, since what applies to these applies
likewise to each of them in composite distributions — ^we would
find certain salient points which must be noted about the
independent variables in each type.. Thus in describing (or
cataloging) an abstract distribution the important things to
note are the basis of the distribution (whether it is nature of
diseases, causes of accidents, kinds of articles, races of the
population, sex, marital condition, or what not) and the number
of items in the series. Little more can be done to categorize;
the abstract distribution. And we may note that in graphics,
the abstract distribution is amenable to no connected method
of illustration, such as maps or curves, but is limited to bar-
charts (including 100% bars and circles, or pie diagrams) and
area bars.
In the geographic distribution there are more salient points
to be noted. First, we should note the whole (such as world,
continent, country, state, etc.) and the parts (continents,
countries, state-groups, states, counties, cities, etc.), into'
which the whole is divided or distributed. Moreover, for
convenience we should also note the number of parts, and their
completeness (that is, whether or not their sum forms the
total). Population and sales statistics are often complete,
building statistics, and morbidity and mortality reports are
examples of commonly incomplete data (compiled from only a
few states or cities). In the graphic presentation of geographic
distributions we can use maps in addition to the bar-charts
which are applicable to all types of statistics, but we cannot
use curves. For complete data we can generally shade or color
whole areas on the map, but for incomplete data, such as re-
ports for various cities, we should use isolated points on the
map.
In the frequency series we need not only to know the basis
of the numerical relations of the series and the number of items
in the series, but we need also to know something very much
THE STATISTICAL MATERIALS
677
akin to completeness, namely its continuity. By continuity
is meant whether the independent variable be a discrete or a
continuous series, and if continuous, whether it be “point” or
“period” data. By point data we mean data for separated
non-contiguous points in the range; by period data we mean
data for connected contiguous or overlapping periods in the
range. For all these forms we can use curves, as you know,
in addition to the ubiquitous bar-charts. But for the discrete
series, generally composed of integral varieties, we are limited
to the staircase or rectilinear curve so closely akin to bar-
charts. Indeed this type of distribution has often little more
than the accident of numerical designations to distinguish it
from the abstract distribution. For the continuous series,
generally of graduated variates, we should give a truer picture
by using the smoothed curve or frequency polygon, though
for period data we make a sacrifice of the accuracy of area
representations thereby. And we may note a further distinc-
tion that while for discrete and period-data continuous series
we should plot the data in the spaces between ordinates, for
point-data continuous series we should plot the data on the
ordinates. These distinctions have been discussed in the
chapter on amount-of-change frequency curves.
Lastly we come to the historical series. Here four salient
features of the independent variable should be noted. There
IS the range of time (the whole) covered, the intervals of time
(or parts) used, and hence the number of items, and finally
their regularity and continuity. The range and intervals are
in centuries, decades, years, months, weeks, days, hours, and
SO forth. And some tables use dilFerent intervals in the same
distribution, such as a series of decades followed by the indi-
vidual recent years and lastly the most recent months, making
in all for irregular intervals. The continuity of the data here
means simply a point and period distinction. Items for
isolated points of time, such as stock or balance reports at the
first of each month, or price quotations at the end of each
week, are point to point data. These the economists call
stock or fund figures. Totals for periods of time, such as pro-
duction or shipment statistics, are period data. These the
economists call stream or flow figures. And it is of course
possible to have isolated periods reported without the inter-
vening periods. All historical data can be presented on curves,
as well as bars, but in the summary chart the period data is
CUJRTS GRAPHS
678
shown by bars, the point data by curves. In general, wherever
we wish such a refinement, we can perhaps more accurately
show period data by staircase curves or vertical bars, and point
data by smoothed curves. A far more frequent distinction,
however, is that while period data can be more accurately
plotted in spaces between ordinates, point data can only be
plotted upon the ordinates. These considerations have been
brought out in various portions of the text.
To the foregoing discussion of statistics as regards the
independent variable, we have now to add a brief outline of
what may happen to the dependent variable. And here we
can no further escape another general distinction which can
be made between what might be called primary and secondary
or derived statistics. The primary statistics consist of totals
or averages which represent the original observations reported
in the statistical table. Now these totals or averages, which,
by the way, form the dependent or adjectival variable in the
table, can be subjected to statistical treatment and materially
modified, and the statistics which have suffered such treatment
may be called secondary or derived data.
There are two main kinds or processes of statistical treat-
ment which can take place simultaneously or individually.
For the sake of simplicity they may be called compilation and
conversion. Statistical compilation is to some extent operative
upon both the independent and the dependent variables."
Statistical conversion is almost entirely limited to the depend-
ent variable and though it is perhaps considerably the more
intricate subject, will receive scant attention from us, as it
does not greatly affect the charting method.
Statistical compilation begins with material in its crude
form, which is a mere listing or list. In the case of the logical
distributions this is also about where it ends. Much rearrange-
ment is possible, of course, but the abstract and geographical
distributions always remain nothing more than lists. This,
perhaps, is why neither can be shown in any graphic form
where connection-lines represent continuity or sequence, in
short, in any form of curve. The numerical distributions,
however, can be so arranged that the items follow each other
seriatim in a sensible way. In the early chapters of the book
we have spoken of this as a case where the stubs or independent
variable facts fall into an order imposed by themselves.
. THE STATISTICAL MATERIALS 679
Obviously it is nothing more than mere mathematical sequence
in the stubs, which dictates this order.
When a numerical distribution has been arranged in this
ordarly way it ceases to be a crude list and becomes a special-
ized one which is commonly called a “series.” It is now usually
ready for charting in the form of a curve, for there has appeared
in the data a thread of connection running through the various
items, a thread which enables us to connect the items on the
chart by a line. This curved line can be shown on scales of all
the various types discussed in the sections on curve-charts,
from simple arithmetic or amount-of-change scales to logarith-
mic, rate-of-change, and other projections. It is not to be
thought that the curve is something radically dilFerent from
the bar-chart; indeed, as you know, the amount-of-change
curve is simply a convenient sort of short-hand symbol of a
series of bars, and the rate-of-change curves are merely special
warpings and distortions of the amount-of-change curve with
intent to bring out hidden relations and features in the curves.
But it remains true that in the curve chart we are really for
the first time, able to shift the focus of our attention from the
comparisons or changes between individual items in a distri-
bution, to something more complicated, the comparison or
changes between these changes in different parts of the same
distribution or in the same parts of different distributions.
And to make the curve, we must first compile the numerical
list into a numerical series.
This first step in statistical compilation is important,
because to the casual reader it is hardly apparent. Indeed, it
may be that the layman, glancing over a volume of the census,
or over any other statistical series, is quite often under the
innocent impression that the figures he sees just grew, some-
what as did Topsy. He does not suspect that many days or
months of study may have gone into the determination of the
proper group or interval limits in the series, and that over a
year thereafter whole batteries of clerks and computing
machines have sorted and enumerated the items in accordance
thereto. Not all compilations are the result of such great
attention. Regrettable it is to say, that publications still
occur in which the raw material, the crude list, is given; but
the compilation of data into series has not been carried far
enough. Such lists and imperfect series tabulations are very
likely to puzzle the student unless he detects the unfinished
68o
CHARTS AND GRAPHS
treatment and completes the* process of orderly series arrange-
ment.
The simple series is the major step dn the tabulation of
numerical data but it is by no means the last, if the data be
of the ^'period'' type. If the data refer to scattered, isolated
points of time in a historical series or points in a continuous
frequency series, it may not be feasible to subject it to the
processes about to be described. But period data (including
discrete frequency series) in which the periods covered by the
items are co-terminous and contiguous, can be subjected to
the familiar process of cumulation and moving total (and
average) calculation. In effect these processes change the
separate groups or intervals into overlapping groups. In the
cumulation, the overlapping is in one direction only; in the
moving total (or average) (taken in its proper position as at
the middle of its period) the overlapping is in both directions.
Slight differences may be observed in the susceptibility of
the two numerical distributions to these processes of over-
lapping. Thus the frequency series can be cumulated in either
direction, either backward or forward, yielding a ^‘more than”
or a ^^\ess than” cumulative. It may then be plotted in the
familiar form of the ogive curve, of which both axes may be
either arithmetically or logarithmically projected and for
which the probabilities curve is the great analytical medium.
The historical series, on the other hand, is sensibly cumulated
only in one direction, and the curve of the cumulative is most
especially used in the Zee-chart and its bars in the Gantt
Progress chart. The historical series, moreover, can be sub-
jected to moving total and average calculations, for any length
of time or periodicity, and the moving series is to be used in
curve charts of all kinds, while the frequency series is never
subjected to this process except in some statistically technical
calculations of the mode and smoothing processes. The subject
of moving totals and averages and cumulations has been dis-
cussed in detail in the text.i
We come, lastly, to the other form of statistical treatment
by which secondary data can be derived from primary stat-
istical sources. It relates normally to the dependent variable
and is the process of conversion of absolute data into relative
1 Beyond the cumulative and^ moving or progressive totals, which are somewhat
in the nature of integrals within limits, the processes of differentiation and integration
are not included in this discussion.
. THE STATISTICAL MATERIALS
68 1
data of various kinds. The absolute data is the original data
Itself, whether statistically compiled or not. It occurs in the
form of totals or averages, which measure items in terms of
various units of volume or value. The relative data is always
the result of comparing this absolute data with other absolute
data. The latter, with which comparison is made, may in
general be called the base of the relative data. The relative
data itself is of several varieties, which we will briefly mention,
and occurs under many different names. Its relativity is
usually obvious enough but is sometimes so completely dis-
guised by ambiguous titles or nomenclature as to be difficult
of detection and when this occurs its analysis may prove a
baffling problem to the inexperienced.
The most familiar form of relative data is the percentage,
or series of percentages, of which the total is 100%. It is the
result of comparing the parts to the whole, or the items in a
list or table to the sum thereof. In particular we may note
that the base is common and constant, the same base being
used for each and eveiy percentage. Hence the percentages
bear the same relation to each other as the original numbers
or quantities which they represent bear to each other. The
percentages, in fact, are but the same statistical facts reported
in terms of a new unit of measurement. And so the percentage
series may.be subjected to cumulative and moving total (and
•average) calculation almost as readily as the original numerical
quantities.
The next important type of relative data is that to which
the special name “relative figures” is given. These share with
the percentage figures the feature of a common and constant
base; but differ from them in the relation to their base.
Relative figures are not to their base-figure as parts to a whole;
their sum does not total one hundred per cent. The relation of
relative figures to their base is an item to item relation; that
is, it is the result of comparing the original numbers, sometimes
called by distinction, the “numerical data,” with one of the
component items. When the base-figure is more or less ima-
ginary, being a combination of several often incommensurable
base-figures, the relative figures are called “index numbers,”
though the latter term is by some writers loosely applied to all
relative figures. Owing to the use of common and constant
bases (for each series), relative figures and index numbers can
CHARTS AND GRAPHS
682
be subjected to moving total and average calculation, but
their cumulation is usually of no value and significance.^
The last group of relative data differs from the foregoing
in the use of various and different bases within each -series
instead of a common and constant base. Such series are always
formed by comparing the items in one series with corresponding
items in another series. Hence the items in the second series
are the bases for the items in the resulting relative data.
Where the latter stand in the relation to their bases of parts
to wholes, they form percentages. Where they are not in this
relation, the most common result is a per capita figure, '^per
family,'’ ^^per dealer,” etc. In either case when the fraction of
ratio is very small it is usual to multiply by a thousand or some
other constant, and so achieve a ^^rate.” These relatives come
in a wide variety of ways and under an equally wide variety
of names. But they all have in common the feature of shifting
inconstant bases. And as a result they cannot ordinarily be
cumulated or smoothed by the moving total (or average)
process. When we desire to compile them in these ways it is
only proper to return to the original data and the base-figures
' and perform the operations upon them, that the smoothed
relatives may be secured from their comparison.
At this point we draw to a close a very rapid survey of
types and varieties of statistics as such. It is our hope that
the reader will find such a panoramic view of value in his
statistical work and the charting problems that arise therefrom.
It is our hope that in time the classification and cataloging of
statistical forms will become so simplified and improved that
it can be used with immediate profit by the novice, as floral
keys guide the amateur botanist to the name and description
of the wayside flower. There is no reason why this codification
and systematizing cannot take place, the structure of forms is
really very simple, and the writer has no patience with the
pernicious, though often unconscious, attempt to throw dust
- in the eyes of the layman and make technical problems appear
more difficult than they really are.
He who has read with a broad comprehension the text to
which this survey of statistical forms is the conclusion, will
® A minor variation of both this and the next type of relative, is the chain-per-
centage or link-relative, in which the base is not constant, but is always the preceding
figure in the same series. It is a form of differential or successive differences. See
rote in Chapter XXVI, page 307.
, THE STATISTICAL MATERIALS
683
understand that graphic forms as well, are varieties and varia-
tions of a common root illustration. He will know that this
common root pictur-e is the representation of a single number
by a straight line. He will know that any collection of numbers,
be it an abstract, geographical, or numerical distribution, can
be presented graphically by a collection of straight lines,
which, if joined end to end, form a 100% bar; but if placed
side by side form a bar-chart. The development of the pie-
chart ftom the 100% bar, he will understand as merely a sub-
stitution of the circular for the straight line. The development
of the curve he will understand as merely the connection and
epitomizing of the bar-chart, suitable only for numerical series,
whether frequency or historical. The varieties and modifica-
tions of curves will have no mystery for him. The area and
three-dimension charts will stand before him as amplifications
and combinations of bar-charts and curve-charts, suitable for
interlocking composite distributions. The map will be but a
variant of the latter, in which two dimensions of the paper
picture the independent variable in geographic distributions,
by longitudes and latitudes. The 100% triangle, the nomo-
graph, and the calculating charts will be but patterns in which
the two dimensions of the paper are devoted to the laborious
illustration and proof of the very simplest propositions in
gebmetry. This is really all there is to charts.
Chapter LViH
THE FUNCTION OF CHARTS
A world turning to a saner and richer civilization will be a
world turning to charts. From this conclusion, unwelcome as
it may be, there is no escape. The case for the chart may even
be sketched in a few schoolboy syllogisms, woven through the
related ideas; civilization, clean-cut thinking; precision of
thought, numerical statements; statistics, charts. With the
last step in this chain, this book has attempted to deal. With
a brief summary of statistical data the last chapter has provided
us. There is no need to dwell upon the importance of precise,
clear thinking, either in business or in economic studies.
■ It remains to glance ahead a bit at the mechanics of the rela-
tions which charts will assume with the civilized world at
large, and to venture a few predictions as to the nature of
these relations.
And for this larger view it seems well to begin by amplifying
our original definition. A chart is an image or graphic repre-
sentation of abstract relations. Where these relations are not
of a numerical nature, the chart is non-mathematical in
character and is closely akin to the other graphic arts of a
purely pictorial character; indeed its only distinction from
paintings, photographs, and the like, appears to lie in the
abstract nature of the ideas which it diagrammatically or
schematically expresses. But where the relations are numerical
and the subject of the chart is statistical, the chart is mathe-
matical in character and forms a distinctly new branch of the
graphic arts. While the artist will seek to present two groups
of ten and twenty horses' each by a picture of so many horses,
placing his emphasis upon the realistic likeness of his drawing
to horses, the chart-maker will seek to present the same
objects by, let us say, two bars, which by their lengths express
the numbers twenty and ten. His chart of horses will be
exactly like his chart of two similar groups of ships, or his
684
THE FUNCTION OF CHARTS 685
chart of two very much larger groups of horses in which the
group proportions are unchanged. He can, indeed, with equal
facility make a chart-for groups of two million and one million
horses, a task which would be beyond the powers of the artist.
In subject-matter, then, the chart is universal, and hence,
too, in its potential appeal and usefulness. No one can think
of two numbers and attempt to comprehend their significance
without, at least unconsciously, visualizing them; the number
which does not conjure up in our minds some picture of quan-
tity remains meaningless to us.
In this sense, therefore, everyone who deals with numbers
is already a chart-maker and a chart-user. We have no choice
between the use of charts and the use of statistics; we have only
a choice between the use of written or physical charts and the
u.se of imagined or “mind’s-eye” charts. Often, indeed, the
latter are sufficient, and many persons, it is true, still prefer
under all circumstances to carry all the pictures of their num-
erical data in their minds. But for the careful study of import-
ant figures, or for the casual study of large bodies of important
figures, this is obviously the less efficient method, and the
physical record, the written or graphic chart, comes into
service. It is more permanent, more convenient, and more
accurate.
The technique of the chart is also, in a sense, wider than
that of the other graphic arts; indeed, it comprises something
of the technique of all the arts. The reader of this book has
seen that we have drawn statistics with pictures, and sculpted
them as models and we have reproduced them by photo-
graphy and by lantern slides and by printing. In this we
have freely used design, relief and color. It may not be too
much to add that some day we shall set charts to music, to
enhance their graphic value, evolving a musical expression
of statistics. This will seem less improbable when we consider
its use in the accompaniment of moving pictures of charts.
The animated chart, made possible by the motion-picture
film, has long been a dream of the author. Its graphic value
will be great in the presentation of fundamental economic facts
to the general public, or of special statistics to special audiences.
By its means the important chart can be presented in various
stages of completion, and attention can be focused in turn
upon each change, development, or addition to the picture.
Thus in a bar-chart, the labels can appear first, then each bar.
686
CHARTS AND GRAPHS
with its data, can appear, one after the other, until the bar-
chart is completed. Curves can be shown wiggling across
co-ordinate rulings, with close-ups of each important added
wiggle. Maps can appear first in outline and the shadings
can appear and spread across the map by simple tricks of
photography, and these shadings can be altered to show
changing conditions for successive points or periods of time.
The ‘^movie’*’ of statistics is clearly coming, for schools and
colleges, for the general public, for the scientific or academic
meeting, and in business, for director’s meetings, for sales
conventions, and for advertising purposes.
In all chart-making, a distinction which will become in-
creasingly recognized is the distinction between charts for
popular consumption and charts for research purposes. This
is no more than adapting the chart to the audience for which
it is intended. And there can be as many diflFerent proper
charting ways as there are different degrees of familiarity with
charts and ease in chart-reading. For extremely popular
presentation, the pie-chart is always effective; bar-charts
should be converted into series of circles and curves into
vertical bars whenever possible. For more sophisticated
readers the amount-of-change curve can be used; for the tech-
nical and semi-technical, the simpler forms of the rate-of-
change curve are permissible. The probabilities and other
special projections will be really understood only by the
experts; and are essentially charts for internal consumption
in the research laboratory.
Though everyone can be told how a bow is carried across a
violin string, we do not expect all to play the violin well.
And though the technique of chart-making can be simply
explained, we cannot expect everyone to make good charts.
The chief source of good charts will always be the statistical
departments of large organizations. When the organization
is an institution for the promotion of research in some special
field, the statistical staff will of course be well manned. But
the greatest strides, at least in chart-making, if not also in
statistical methods, will in the future be made in the statistical
departments of large business organizations.
In business the function of the statistician is two-fold,
comprising on the one hand special research and investigations,
and on the other hand, the co-ordination and intelligent report-
ing of current business operations. In both of these, charts
THE FUHCTIOH OF CHARTS 687
are essential implements. In the research field the statistician
has often a scouting function, his job being to look ahead and
try to forecast the future development of the house and its
markets. In the reporting field he assembles and interprets
the operations of all the other departments; purchasing, pro-
duction, shipment, warehousing, sales, and other collections;
and of the business as a whole: inventories, costs, and profits.
His position here is that of liason or intelligence dfficer between
the responsible head of the business and his subordinates*, and
also between the responsible subordinates and their depart-
ments. To get the fullest use of the expert intelligence in
visualization and analysis, one of the vice-presidents may be
himself a professional statistician and chart-maker of the
highest specialized training, but in the past the average statis-
tician has not often displayed a sufficiently practical view-
point to justify this connection and the wealth of significance
which lies in the records of the individual business house is
untapped by those who must guide it.
Comparable to the lawyer who brings to the guidance of
business enterprise an intimate knowledge of legal technical-
ities, is the business statistician who brings to it an intimate *
knowledge of statistical interpretation. In business houses
where the operations and problems are of astandardized nature,
his' skill will not, except in very large concerns, be constantly
needed and the statistician here becomes a consulting expert
rather than a permanent officer. In such concerns the report-
ing procedure can be quickly set in motion and standardized,
so that it can be carried on thereafter by clerks. The Gantt
progress-charts and a few of the simpler curves and maps are
all that need be installed, after the proper system of records
from the accounting and other departments have been estab-
lished. In business concerns of more variety of operations,
the trained statistician is necessarily more of a permanent
member of the personnel and the work of forecasting is likely
to be seriously entered into. Here the widest variety of charts
come into use, for a nice understanding of their graphic value
and true significance is available. Here it is often profitable
to maintain a special statistical '‘laboratory’' with complete
facilities for statistical sources, compilation and analysis and
for graphic records.
The well-furnished statistical department should, of course,
contain the mechanical calculators, the double-entry adding
688
CHARTS AND GRAPHS
xy
Computihs Clerks
O
or
Typist
D
PLOOR-PLAH
for
Small
Statistical Department
KEY
P * Projector
C s Calculating machine
A = Adding and listing machine
T ~ Typewriter, one long carriage
one variable typ
D = Desk
L = Light -box
P « File
S • Shelves
ft * Map and tracings file
Fig. 496.
and listing machines, the multiplying and dividing machines,
and perhaps the card-punching multiple-entry machines (the
true posting machines). It should contain full drafting
facilities and perhaps also blue-printing or photostating ma-
chinery. These things obviously belong to the workroom,
which should be apart from the office of the directing and
THE FUNCTION OF CHARTS
689
creative statistical officer. But tlie department is not complete
without full accommodations for the successful study of its
results. There should be a conference room, convenient to
and properly fitted for the use which will be made of it by the
directors or vice-presidents and responsible heads of the
business.
In the conference room the charts perform their chief-
function in business, as guides to the formation of policies.
The room should be equipped with a light-box for the com-
parison of curves and with a screen and projector for charts
which it is desirable to exhibit to several persons at once.
Important data can be permanently posted on large wall-
boards and these wall-boards, by the use of sliding panels,
can hold large bead-maps as well. All important data should
be on record in chart form, either in looseleaf binders or vertical
files. Needless to say, the room should always be locked up
when not in use and the keys to it should be in the hands of
but two or three persons. It should be the repository for all
information about the concern which is of value in the forma-
tion of policies, this information being in chart form because ,
of the ease with which it can then be consulted.
Of a much more general nature are charts for popular con-
surnption. These are appearing with increased frequency in
.newspapers, general magazines, and technical publications.
The day will come when no statistical compilation will be
regarded as complete until it is illustrated with charts which
present its major significance. The greatest development of
charts, here, however, will take place in the advertising
columns, and in general for propaganda work. For the proper
chart is an excellent weapon against the inertia, indifference,
and often hostile attitude of the average reader. It is not
mereiy the best kind of eye-catcher for calling attention to
numerical data, it is also the most convincing proof of that
data. The most casual reader stops a moment before any
diagrammatic puzzle to examine it. If he finds incidentally
that he immediately understands it, he is perhaps at once
pleased with it and is sure at least to carry away with him a
memory of the message it conveyed. That such charts should
be of the simplest, goes without saying; and here too, expert
skill in chart-making is desirable. For the right chart is
strong in inverse ratio to the technical ability of the reader,
690
CHARTS AND GRAPHS
and the less effective would be text or tables, the more powerful
grows the right chart.
In all fields, scientific, academic, and commercial, the chart
is a medium of expression too forceful to be overlooked, too
valuable to be neglected. Its future growth will assuredly be
rapid and perhaps in many ways even startling. In this book
we have set forth many ways for the presentation of statistics
and statistical relations. The category is, however, by no
means complete. It cannot be complete, for the charts are
still in the making, and the methodology of graphic illustration
is in no sense that of a perfected art. There is room for much
improvement in existing chart-forms as well as in the develop-
ment of altogether novel forms. New ideas will come out of
the research laboratories, new methods, new forms, new
charts. The distinction between graphs for popular publica-
tion to the general public and graphs for internal consumption
in the statistical workshop, will become more marked; and as
public knowledge increases, charts will pass out of the work-
shop into the magazine and book page, no less through adver-
tising than through text columns.
We are finding a new language, the grammar of which is
not yet completed, nor the dictionary written. It is well that
this is so, for codification and systemization easily bring
stagnation; and volumes such as the present, in which the
existing material is set in order, must not be allowed to stifle
new growth. The reader is urged not to permit the rules laid
down in this book to restrict his efforts, but rather to allow the
principles set forth to stimulate his imagination and enterprise.
The pictorial display of mathematical and numerical state-
ments is an illustrative art, with the high object of facilitating
human understanding and vision, an end the achievement of
which justifies all means, be they orthodox and accepted or
novel and previously untried.
FINIS
APPENDICES
Appendix A
IMPLEMENTS FOR MAKING CHARTS
The equipment which should be available in a chart-
making or statistical office depends, of course, a great deal
upon the types and forms of charts which will be developed
and the nature of the data which will be handled statistically.
Charts can be made at home or in the very smallest office,
with nothing more than a draftsman’s ruling pen, some India
ink, and good paper. As the chart-making work grows, more
drawing pens will be added in order that different colored inks
can be ruled in without delays, and in order that several oper-
ators can be drawing at the same time. A good drawing board
is necessary and perhaps two drawing boards are best, one
about 24 by 30 inches for large charts, and the other about
18 by 24 inches for smaller charts. Together with the drawing
board, there should be a T-square and small triangles or tri-
squares. Occasional need arises for one or two patterns of
French curves. A good drawing set includes a compass or
dividers for the drawing or circles of circular outlines. Several
dotting machines for the drawing of dotted, broken, and dotted
dash lines are on the market, and when they can be success-
fully used, save considerable time, but they are not always
successfully used. A section-liner is almost essential for much
cross-hatching work. A protractor is necessary where circles
will be divided into proportional parts and angles must be
used. For bar-charts, a double ruling pen, or railroad pen, is
a great convenience, adding to the appearance of the chart
by making more absolute the uniformity of the bar widths,
and greatly decreasing the amount of time required for the
making of the bar-chart. (This pen rules in two parallel lines
simultaneously.) Bar-charts can also be made on large scales
with adhesive tape or passe-partout; and several mechanical
bar-charts are now on the market in which cloth tape is un-
wound from invisible or hidden spools and drawn out to the
691
CHARTS AND GRAPHS
6y'2
required length of the bar-chart. These last are useful where
the length of the bar-chart must be frequently changed and
brought up to date (as in a sales-manager’s of&ce where the
bars represent the weekly averages or cumulatives of the^work
of the individual salesmen). In map work, a curved ruling pen
is sometimes an advantage for the drawing of rounded curves;
better still, is the Payzant lettering pen. A planimeter is
Permission of Keuffel b* Esser, N. F.
Fig. 497. The Payzant Lettering Pen.
often useful for checking up total areas on the map. A panto-
graph is a device by which outlines can be copied on larger or
smaller scales, and is sometimes useful in map work; very
cheap pantographs can be obtained, which will ordinarily be
satisfactory.
A straight-edge is useful for the cutting of paper, and the
best knives available are the ordinary one-sided razor blades,
as the paper-cutting knives require frequent sharpening. The
straight-edge is necessary because from time to time in the
cutting and trimming of paper the edge itself will be cut into
by the knife and if the T-square has been used the T-square
will then be ruined. Ordinary camel’s-hair brushes are often
used for applying water-color or ink to maps; but the best
way of coloring maps, as has been previously described, is
by the use of wax crayons, the cheaper and waxier the better,
the wax being afterward removed by a sharp knife-edge or
razor blade, leaving the desired color tint. For the quick
filling in of bars and solid areas, the lettering pen is desirable.
It can be obtained in many sizes, but the results are not
entirely even and smooth unless the operator is skilled. A
special pen (the Payzant) has been put out for fine lettering.
IMPLEMENTS FOR 'MAKING CHARTS 693
which has a round nub with a well similar to that in a drawing-
pen, holding considerable ink. This pen is also available in
several sizes.
Typewriters should be used as far a® possible in the letter-
ing of charts, as well as in the entry of data upon the charts.
The process is much more rapid than hand-lettering, and the
results are, for the average-sized chart, usually better. There
are two sizes of standard typewriting type, “pica” (10 char-
acters horizontally to the inch) and “elite” (12 characters
horizontally to the inch). Unless large type is especially
desired to facilitate small reproductions, the elite is better,
for it enters all figures in less space. Any special type-faces,
such as Gothic or Italic, may be had, but the usual type-face
is Roman. The figures come in two styles in all machines,
“book-keeping” and “regular.” Book-keeping type has
slightly greater visibility but gives uneven lines, as the
numerals have swinging tails. The regular numerals are
usually more satisfactory.
The typewriter carriage to use for chart-making should
accommodate 11-inch paper (and larger, if charts are being
made on sheets larger than 8|xll inches). One standard
machine (the Royal) will take 11-inch paper on its regular
carriage, all other makes require long carriages. The type-
writer to use for chart-making should also so hold the paper
as to print down to the very bottom edge of the paper without
shifting. There is but. one standard machine (the Royal)
which will do this. Besides the standard machines, the
Hammond has good features for chart-making in that it will
print any style of type at a moment’s notice, 9 lines to the
inch instead of 6, and will space the characters properly (and
if desired, as close together horizontally as 18 characters to
the inch); but this machine is not so easily handled by most
operators as the standard machines, because it has a three-
shift key-board, and, while it will accommodate any size of
paper, the paper is likely to shift slightly in it and cannot be
used down to the bottom edge.
For the statistical work, special computing machinery is
desirable, both for its speed and for its accuracy, being for these
reasons, where the amount of statistical work to be done is
considerable, a great economy. Computing machinery is in
general of two types, the adding machine and the calculating
or multiplying machine. The adding machine can be obtained
694
CHARTS AND GRAPHS
in both printing (or “listing”) and non-printing types. Ob-
viously, the printing machine is far more desirable because the
operation can be checked back by an examination of the printed
page or record. A special type of adding machine, called the
duplex model (Burroughs), is desirable for work in which
totals of parts will be required. These part totals are known
as transfer-totals, the machine having two faces, the one of
which clears without removing the record from the other face,
so that several adding operations can be conducted at the same
time. The duplex machines are particularly useful in the com-
piling of statistics by States, the part totals being taken off
for State groups, and afford a great saving in the detection of
errors when the data is checked over. Still a third type of
adding machine is the mechanical tabulator (such as the
Hollerith), which works with punched cards in which the
amounts to be entered with full descriptive detail, are repre-
sented by holes in a card, and these holes operate the machine,
just as do the holes in the records of the player-piano and
similar devices. This is the true posting machine — it sorts,
posts, and adds, with a typewritten record if desired, auto-
matically.
The calculating or multiplying machines so far manufactured
are only of the non-print type and leave no written record. This,
of course makes errors more difficult to detect. There are two
general types of machines. The first (such as the Burroughs or.
Comptometer) automatically adds as fast as the operator punches
the keyboard, and calls for specially trained operators who can
punch several keys at once and continue punching the same
keys the proper number of times to effect the multiplication
(shifting the position of their fingers on the keyboard for
each new digit in the multiplier). The second type of machine
performs the same operation automatically from a single
punching of the keyboard. The operator may be required to
turn a handle the proper number of times to effect a multipli-
cation, and to shift the recording dials one space for each digit,
but the work is very quick and absolutely accurate. In the
latest model German machines, electrically driven, the work
is entirely automatic after the setting of the keyboard. Adding
and calculating machines are economical of time and expense
if electrically driven.
All of these calculating machines are virtually multiple-
adding machines, for effective quick addition. They are
IMPLEMENTS FOR- MAKING CHARTS 695
accurate to the last figure recorded. For many statistical
purposes, however, such as the figuring of percentages, accur-
acy is not necessary beyond the third or fourth figure. For
such,, work, a slide-rule is sufficient and, though harder on the
eyes, is much more portable and is indispensable in the statis-
tical office. The accuracy of reading increases with the length
of the rule; and slide-rules are made five, ten, and twenty
inches long. Further accuracy can be gained ’by the use of
magnifying glasses fitted on the runners of the rule. Needless
to say, slide-rules will perform many operations outside of the
powers of the calculating machines.
Appendix B
STEPS IN MAKING CHARTS
In statistical oiHces, both large and small, it is desirable
to have as near as possible an approach to what may be called
'‘straight-line methods’’ of chart production. Only by the
institution of such methods can the routine work of a large
number of charts be satisfactorily and economically accom-
plished. For this purpose, it is desirable to break up the work
of chart-making into various steps and stages, and to have the
individual charts, as they pass through these stages, pass from
the desk of one operator to another in a regular series and
direction. The various steps outlined below will be found to
be a fairly complete list of the stages through which different
kinds of work will pass. Very often, however, some one or
more of these stages will be omitted for particular charts and
for particular data,
I. The first step in chart-making is, of course, the collec-
tion of data or statistics, namely, the information to be shown
upon the chart. This information can be gathered in two dif-
ferent ways. The first way ought, wherever possible, to be
followed out, whether the second is used or not.
^""The first way to gather data is to consult and collect all
information previously compiled by other investigators on the
particular subject. This is a class of research work ordinarily
involving the consultation of the various books, pamphlets,
and other authorities in the public libraries, and calling for
the services of fairly skilled library workers. A reasonably
complete knowledge of the various sources and authorities in
which the particular information sought is likely to be found
in its most useful and complete form, is of course desirable,
in order that the search will not take too much time. When
the information has been found, it can be carefully copied
upon specially prepared work-sheets or data sheets by the
investigator, or, if the data is in compact form, it can
696
STEPS IN MAKING CHARTS 697
be photographed or photostated and the photostatic copies
used in the office. The latter method, though apparently
more expensive, is far more accurate and reliable and more
economical of time, so that, in the end, it is generally the
cheaper process.
The second method of gathering the information is by the
use of field investigations. It is an independent and original
study for the purpose of securing primary information rather
than secondary information (i.e., information taken second-
hand from other investigators). The field investigation may
be carried out by a skilled investigator, if it is not too exten-
sive and if the resources are available for sending a thoroughly
skilled investigator out. When this is not possible, or when
the extent of the investigation is very great, the method of
questionnaires can be used. In this case, the art of question-
naire-making comes into play, for the drawing up of a question-
naire is by no means as simple as it might appear.
Z"'" A good questionnaire is one in which no question can be
misunderstood; one in which each question is capable of only
one meaning and of a precise answer of one type only. More-
over, a person who has drawn up a questionnaire must be
able, beforehand, to envisage his entire problem, foreseeing
all moot points and issues which will arise in the course of the
inv'estigation, and to which answers will be desired. Lastly,’
the questions must be so framed as to avoid, as far as possible,
any psychological reactions either upon the part of the investi-
gator or of those whom he questions and consults for his in-
formation. In fact, the psychological difficulties about many
questionnaire problems are the principal obstacles, and the
method of questionnaires is fast losing ground for precise
statistical compilation, because of the careful analysis and
psychological interpretation, translation, and correction to
which the answers must be subjected before they can be satis-
factorily compiled. And great as is the task of preparing and
conducting a satisfactory questionnaire, the problem of
compiling its answers is sometimes even greater, and requires
a staff of more than ordinary intelligence.
And in both research and field investigation work, wherever
clerical tasks have been performed, it is, of course, necessary
that careful checking be done on all such clerical work in order
to catch and correct errors which are humanly inevitable. A
definite place in the schedule of preparing charts must be
698
CHARTS AND GRAPHS
given, to this work of checking back for accuracy on all the
clerical work performed.
n. The second step in the routine, of chart-making is
ordinarily called the computing. This often requires that the
data be first copied upon specially prepared forms or work-
sheets, so that it may be subjected to the proper processes.
The computing should, as far as possible, be planned con-
siderably in advance in order that work may progress evenly
and smoothly. The work is of a clerical or statistical nature
and calls for the services of statistical or computing clerks, and
frequently also for a battery of adding and calculating
machines. The work-sheets which are to be used for the job
should be carefully designed with an eye to the machinery by
which the processes of calculation can be most easily performed.
Thus, in cases where totals and subtotals are desired, the work-
sheets should be designed so as to fit into the adding-and-
listing machines in order that the machine may operate
directly on the work-sheet and not upon the usual tape. If
the work is first done on the tape, it will have to be copied on
the work-sheet, with the unavoidable percentage of error and
'the great additional time and labor involved. The only alter-
native is to paste the tape on the work-sheet, and this will
not be possible if the work-sheet is not large enough. Needless
to say, for all computing steps the most rigorous checking for
errors is necessary. If possible, the computing should be so
done that it is self-checking, or easily checked for accuracy by
a single checking operation performed upon the totals for the
data. It is best, therefore, when the adding machines are to
be used, to have work sheets in which the lines coincide with
the adding machine lines, so that the sheet can be run through
the machine instead of tape. Moreover, if possible, the
items to be added should be entered in a column, with blank
lines between them, so that the machine entries may appear
immediately below the hand-written entries, this reduces error
and facilitates checking.
in. The next step is the beginning .of the chart-making.
The general character of the charts to be used should have
already been determined. Suitable chart-forms should have
been obtained from some publisher, or made to order by one’s
own printer. These forms should accommodate the data in
typewriting. And this step, therefore, may be called “entering
up the chart. It calls for the services of an intelligent and
STEPS IN MAKING CHARTS
699
capable ''tabulating typist/' for the work must be both accu-
rately and neatly done. If, in the case, for example, of curve-
charts, the ordinates, at which the data must be entered have
been^ placed at uniform typewriter intervals of one-third of
one inch, the typist will be able to work rapidly and smoothly,
and work will cross this desk promptly. One good typist,
under such circumstances, can keep half a dozen compiling and
drafting clerks busy, and will generally find time to do comput-
ing work as well. Again, checking for errors is necessary.
IV. When the final data has been entered upon the chart,
the next step is one of plotting this data in chart form. The
immediate proximity of the data to be charted upon the chart
itself, as placed there by the last operation, makes this drafting
or plotting process extremely easy, and where the chart-fields
have been already printed or drawn upon the paper, the
drafting requires little more than the proper selection of
plotting points and the careful ruling in of curved lines or bars.
The work calls for the services of an ordinarily intelligent
clerk and only in the case of veiy complicated charts, or in
cases where extra lettexing and entering of data will be done
by hand, is it necessary that skilled draftsmen be employed.-
The draftsman should work under the best available light, as
the eye-strain of careful plotting or ruling is severe. Art
school students and engineering school students are generally
qualified to perform this work capably. Again, the process of
checking must be carefully done to detect error.
V. The final step on any chart is the labelling and the
finishing up of the various details left unfinished in the type-
writing and plotting stage. Unless the foi'm of title for the
chart has been standardized, the problem of a correct, com-
plete, and easily understood title for a chart is sometimes very
difficult. And the title should, in general, be made by the de-
partment head or statistician who is responsible for the work.
There should always be a portion of the chart sheet in which
the title can be conveniently located where it will be at once
apparent to the reader. In the case of historical curve-charts,
where the field is low on the page, the title naturally belongs at
the top. In other cases where the chart is higher up on the
page, the title can be placed below the chart.
A good title should not merely give the nature of the
phenomena shown by the chart together with the general kind
of analysis followed by the chart, but it should also give such
700
CHARTS AND GRAPHS
distinctive details as will separate the chart from all others in
the series. Where the series of charts is clearly connected,
as in historical curves, the date or year of the individual chart
can be placed in the corner of the paper, the chart title Iseing
reproduced alike on all charts in the series. In addition to
the title of the chart, the chart should also tell its source, that
is, the authority for the information it presents. And, in
addition to these two items which must be typed upon the page,
there is generally considerable labelling of data columns.
Sometimes there is labelling of individual curves upon the
chart-field itself.
When all this has been done and the chart has been dressed
up in its final form, it should be carefully inspected by some
one competent, as far as possible, to detect errors which appear
in the chart; and the chart should have in one corner a place
for the “O. K.” signature of the person in authority who has
finally approved of the chart. In addition to this approval
signature, the data of approval should be shown so that charts
of different date of manufacture can be easily seen and the
latest and most reliable chart distinguished.
Personnel. — In short, it will be seen that the statistical
office has, in the main, five major processes, namely: research
work, computing, typing and printing, drafting, and inspec-
tion (including titling), and, in general, it may be said that
these processes call for different types of workers, namely:
research workers or librarian, statistical clerks, typists or
letterers, draftsmen and artists, and statisticians.
Notes. — In addition to the finished chart itself, it is some-
times desirable to have appended notes or explanatory com-
ments which will serve to interpret the significance of the
chart to the reader or executive who will consult it, and which
will point out to him the important facts displayed by the
chart. The writing and composing of these explanatory notes
should be composed by the statistician in charge. These
explanatory notes should be in the most easily comprehensible
form. Their composition calls for an extremely practical
understanding of the point of view of those who will read and
consult the chart, and requires a return to the language and
to the non-technical line of thought which will be pursued by
the layman.
Reports. — ^Nothing has been said here about the mobilizing
and assembling of a large number of charts upon one subject
STEPS JN MAKING CHARTS
701
into the form of a single coherent report. This is usually con-
sidered a matter for the skill and judgment of the statistician
himself. According, -•however, to so excellent an authority as
Mr. Charles P. Steinmetz,i there are three kinds of reports,
and the most complete report generally contains these three
types within itself. The first is the general report in which
the final conclusions and significances are summarized briefly,
the report perhaps taking up about 10 per cent of the entire
report. It is this part of the report, and often this part only,
which will be read by many executives, or the average reader.
In the second part of the report, these conclusions which ap-
peared in the summary or general report are expanded in
greater detail to show their bases or foundations and to enable
the careful reader to delve deeper into individual phases and
aspects. This second part may contain about 30 per cent of
the total number of pages in the report. The third part of
the report is the technical authority and technical detail
which will be read only by those who are extremely anxious
to check up upon the work of the compiler, either for the sake
of repeating or elaborating the investigation, or for the sake
of detecting errors or confirming the accuracy of the informa-
tion given. This part may often take up the greater portion
of the report.
* Steinmetz, Charles P., Engineering MatheTnatics, McGraw-Hill Book Co, 1917,
p. 290-29J.
Appendix C
METHODS OF PRESENTING CHARTS
In the main, the charts described in this book have been
discussed upon the basis of presentation upon ordinary size
letter paper, that is, paper measuring 8|xll inches. Occa-
sionally, larger sheets of double this size have been mentioned,
and in the section on models the need for mounts and con-
tainers of uniform size has been discussed. The Sfxl 1-inch
paper is perhaps the most generally convenient because of its
conforming to standard sizes of office paper and vertical and
other filing methods. In an office where legal size paper is
used, it would obviously be better to adopt sheets 8|xl3
inches as the standard chart size, and on occasion, to use
sheets of double this dimension.
In the section on curves, which form the great majority
of charts, the desirability of positioning the curve in one
corner of the paper for ready comparison and of leaving large
margins above and to the left of the chart-fields has been
explained. The chart-form itself should be carefully designed
as the one most suitable to the type of chart-work which will
be done.
It is of the greatest importance that the “field,” or rulings,
of the chart-form be in a faint ink, preferably green or gray.
(The orange and reds are hard on the eyes; the blues will not
photograph.) When the charts are to be reproduced on a much
smaller scale, it is well to make all but the more important
co-ordinates in blue, so that they will not be reproduced. For
this purpose, blue co-ordinate paper can be used, the co-
ordinates which should be reproduced being ruled in by hand
in black ink.
The usual types of maps are not ordinarily printed upon
standard sizes of paper, and if a great deal of map-work is to
be done, it is well to have special maps printed upon regular
sizes of paper to conform to the rest of the charts in use. Maps
METHODS OF PRESENTING CHARTS
703
on the S^xll-inch paper, often- of inferior quality, can be
obtained from a few manufacturers of charting materials.
These are sometimes better than more' elaborate maps, as
they *re generally only outline maps showing State or county
boundaries.
Attempts have been made to establish standard charting
forms upon cards for card-catalogue filing, small curve-
charting fields being printed upon 4x6-inch cards’ in one corner
of the card. The fields are ruled ofiF for amount-of-change
curves only, with ordinates for 52 weeks, 12 months, and 31
days, in the usual way for historical curves. The amount of
data which can be entered upon these card charts is, of course,
limited and the detail in which the curves are shown is not
. great. The thickness of the card prevents, to a certain extent,
the facility of “light analysis.” Some publishers present
these small charts on thin paper suitable for tracing or “light
analysis,” as part of a loose-leaf note-book system to be
carried about in one’s pocket.
A major problem in graphic presentation arises when the
charts are to be shown to large audiences. The practice is
often followed of making the drawings extremely large, say
three by four feet in size, on heavy paper which can be unrolled
and pinned against a bulletin board for display. These rolls
of paper, however, are difficult to carry about and are easily
.damaged. This is even more true when heavy card-board is
used, which can not be rolled but must be carried about flat.
The best advice appears to be to present the chart upon
tracing cloth, which can be fastened at one end upon a large
stick of wood and easily rolled or unrolled. Where expense is
no consideration, the window-shade rollers on which school
maps are mounted can be used. These are contained in long
narrow boxes to keep the chart dust-proof, the chart being
unrolled by pulling the lower end out of the box in the same
way that a window shade is drawn down.
Much the best method for the display of charts of large
size to an audience is by the use of lantern slides and a lantern
slide projecting machine. These machines can be purchased
for small sums in a very handy shape, folding up like valises
and easily portable. The lantern slide can be made by any
photographer at small cost, from the original chart used in
the office made in the usual way upon ordinary size paper.
Where colored areas are shown upon the chart, the same colors
704 CHARTS AND GRAPHS
may be shown on the lantern slide, by coloring the slide with
Japanese transparent photographic colors. This coloring work
will be done by the photographer according to instruction, or
according to the original chart which has been photographed.
Lantern slides are smaller than post-cards and easily carried
about. Except through breakage, they are not eiasily
damaged.
The lantern slide method has more to recommend it in the
fact that a drawing which will be seen upon the lantern slide
will easily be visible to the entire audience. Where the very
large original drawings are used instead of lantern slides, the
lines in the drawings have to be made very much thicker to
make them visible, but with lantern slides, the faintest lines,
if distinctly visible upon the plate, will be clearly projected to
the entire audience. A safe rule is that whenever the original
office copy of the chart, from which the lantern slide is made,
is larger than 8|xll inches, the lines upon it will not be clear
and definite upon the lantern slide (because of the reduction
in size) unless the lines are made heavier; but in the case of
originals up to 8|xll inches, not only will the ordinary
markings be clear and distinct but the ordinary typewritten
labels and data will also be visible.
A very expensive machine has been devised, which is
useful in large offices for the display of many charts to board
meetings and other small audiences. It is a projecting ma-
chine which does not require lantern slides, but which will
reflect the image of the original chart upon the screen. The
convenience of this type of machine is, of course, very great,
as it eliminates the delays and expense of lantern slides, and
permits the exhibition of any material at a moment’s notice,
even though the need for such an exhibit had not been fore-
seen. The machine is, therefore, valuable where a large num-
ber of charts may be shown and it is not certain beforehand
which ones will be desired.
The method par excellence for popular audiences is the
moving picture film and machine, with the charts shown as
actually developing and building up. The manufacture of such
films is not easy and is very expensive, but the results fully
justify it. Thus, a bar-chart shown in this way would first
appear merely as a list of items, and then, one by one, the bars
would appear upon the screen, until the entire chart was as-
sembled. Such a chart receives careful study and all its parts
METHODS OF PRESENTING CHHRTS
705
are understood, and their significance is grasped by the audience.
. 1 he reproduction of charts in large numbers involves special
problems. Where only a few copies are desired, photostating
is th^ best method. Blue-printing is a more economical
method But its results are neither so clear nor so attractive.
By the use of blue-printing, only negatives can be obtained,
in which black areas appear as white and white areas appear
as blue. A somewhat similar method is known as the Van
Dyke process, or black-line and brown-line process. These
are obtained partly by offset and partly by photographic
methods, and positives as well as negatives can be secured,
but the original chart should be upon very translucent, un-
water-marked paper (best of all, upon tracing cloth) and very
distinctly and clearly drawn. In these two methods, the
process is a photographic '^printing-through’' one, the light
passing through the chart to a sensitized surface. It is
necessary, therefore, that no extraneous matter be upon the
reverse side of the chart and that no corrections be made upon
the chart by overlaying fresh paper or Chinese white. It is
also desirable that the chart paper be clear, un-water-marked,
translucent. In these respects, the blue or sepia print, and
the black-line or brown-line print are unique. As they are
"printing-through processes,” it is always advantageous, when
there is typewriting upon the original drawing, to back up the
sheet with reversed carbon paper, so as to get an additional and
coinciding imprint of the typewriting upon the reverse side of
the original.
The above limitations and precautions do not apply to
the truly photographic processes, that is, the photostat, the
photograph, and the photo-engraving. In these, the light is
reflected back from the surface of the chart to a sensitized
surface, without passing through the chart. Thick, opaque
paper can be used, with corrections in Chinese white or on
special slips of paper pasted over the incorrect parts; also,
the condition of the back of the chart does not matter. When
only a few copies are needed, photostating is the best process,
far more convenient and only slightly more expensive than
blue-printing. The least expensive course is to get a photo-
static negative and as many blue-print positives as desired.
When suflScient copies are desired to make printing advisable
(that is, printing from metal plate) it is necessary to use either
the "line-cut” or the "half-tone.”
7o6 charts and GRAPHS
The line-cut is the more economical but shows only full
black and white markings. The half-tone (in which the
object is photographed through a screen) is a more expensive
process, giving results with all variations and tints of gt^y as
well as almost full white and full black. Half-tones are much
more expensive than line-cuts and require more time in their
manufacture, but the results, of course, are better when areas
are shaded with different tints and colors in the original chart.
The line-cut is sufficient for the ordinary bar-chart or curve-
chart and can often be given elaborate shadings and cross-
hatchings by the use of the Ben Day process.
For all photographic methods, including printing, care
must be taken in the choice of colors, when colors are used
upon the original chart, for as has been previously pointed
out the various colors reproduce differently by photography
than would be expected from their appearance to the eye,
and certain shades which appear decidedly different to the
eye, may be exactly similar to the camera and reproduce alike.
Red, of course, photographs as black (that is, appears as black
upon the photographic print), while blue does not photograph
at all but appears as white upon the photographic print, and
yellow appears almost black. Thus it will be seen that a scale
from red to blue arranged in chromatic sequence of colors will,
to a certain extent, photograph as a natural sequence of grays
ranging from solid black to white in the photographic print.
Three other methods of reproduction are in ordinary com-
mercial usage, the hectograph, the mimeograph, and the
multigraph. The first two of these can be used satisfactorily
for the reproduction of charts. In the hectograph, the old-
fashioned jelly-offset process, special hectograph ink can be
used in several colors which will simultaneously reproduce,
the copies appearing somewhat paler than the original but
reproducing color for color at a single offset. The number of
copies that can be secured from hectographic offset is, however,
limited. The best offset processes claim to afford as many as
SO to 70 copies from a single original, but as a rule 25 to 30
are all that will be satisfactory. When not more than two or
three dozen copies are desired, the hectographic process is
extremely simple and satisfactory in the average office, its
only requirement being that the special hectographic or copy-
ing ink or pencil, or typewriter ribbon be used in the making
of the original chart which is to be copied.
I^IETHODS OF PRESENTING CHARTS
707
The mimeographic process is. essentially a stencilling one,
the chart being first drawn upon a fine wax or fibre stencil
and then laid over the drum of an inking and printing machine,
the ink passing through the cuts in the stencil and printing
upon paper. The mimeographic process will produce as many
copies as desired up to several hundred, but is limited, like
ordinary printing, to one color only, for the color is deter-
mined by the ink which has been previously placed upon the
drum of the printing machine. The mimeographic process can
be conveniently used where mimeographic reproductions are
already being made of typewritten copy. Its limitations are
that it shows only one color, and that it is impossible to repro-
duce solid black areas on the mimeograph. Shading must be
done by cross-hatching, and extensive cross-hatching is apt
seriously to damage the mimeograph stencil. Moreover, the
mimeographic process may prove a dirty one, even for those
who are experienced with it and certainly for the beginner.
And it is extremely difficult to adjust the stencils upon the
printing drum so accurately and so evenly that the reproduc-
tion will be precise; as a rule, a slight curvature or wrinkling
of the stencil is observable; this, of course, makes for irregular,,
curved, or broken lines on the chart when straight lines are
desired.
• There is a special instrument known as the mimeoscope
which is well adapted to the drawing of charts upon mimeo-
graph stencils. It is also useful in the office as a “light-box”
for general light analysis. The machine consists of a strong
electric light under a ground glass (ground to diffuse the light).
By the use of this machine, charts can be easily traced on
mimeograph stencils, using the various kinds of mimeograph
styles provided for the purpose. The mimeograph method is
adapted to the reproduction of simple charts in one color only,
without solid areas, and not requiring precise reproduction,
when the number of copies desired is between 50 and 500;
for a smaller number of copies, the hectograph processes are
satisfactory, and for a larger number of copies the printing
processes are more economical.
In a few cases where charts are largely prepared upon the
typewriter, and not more than two, three, or four copies are
desired, the carbon copy method of reproduction can be used.
Carbon paper can be obtained in red as well as black, and some-
times in other colors, and the first two or three carbon copies
708 CHARTS AND GRAPHS
are likely to be very good when the paper on which the charts
are made is not too thick or soft. Sometimes the labelling and
typing for a number of copies can be done at one operation on
the typewriter with carbon paper, or in Wo operations when
different colors of carbon paper must be used (first prititing
up all of one color and then substituting the other color of
carbon paper and printing up that). Drafting upon the charts
must be done on the various copies independently with
drawing pens in the usual way, making each of the copies
original so far as the drafting is concerned unless, as in the
case of bar-charts, the chart can be made with the type-
writer. Bound carbon sheets (in binders) help to prevent
shifting of the copies.
Whenever typewriting is done for charts which are to be
blue-printed, or photographically copied by direct printing (in
which case, the light will have to pass through the entire
chart-paper), it is best to insert a piece of carbon paper behind
the original chart and print a reverse copy upon the back of
the paper to add intensity to the typewriting on the chart.
Only fully inked typewriter ribbon should be used for charts
.which are to be photographically copied, as it is desirable that
the printing should be as intense as possible for the best
photographic results.
Appendix D
COLORS IN CHARTS
A great deal of controversy takes place over the use of
colors in charts. In a previous appendix we have discussed
those limitations to the use of colors which arise from the
needs of specified reproductjion processes. In this appendix
will be considered the use of colors in cases where no mechani-
cal limitations are imposed and colors can be judged entirely
upon their own merits.
The argument against colors takes the line that the average
business man is not accustomed to them, and will consider
more carefully a product in the familiar black and white.
The colors in this view tend to make the chart ‘^pretty’’ and
prettiness is rightly to be avoided. Other things being equal,,
artistic effects are always desirable, but never to the point
where they attract attention and distract the reader^s mind
from the message of the chart.
But even in this view, not all lines need be equally strong
and it is freely conceded that the co-ordinates of the field of
a chart should be as light as possible, and, even better than
in black, ruled or printed in gray ink. For the field is only the
background, and the lighter shade throws the curve or plotting
more distinctly into the foreground. And the best practice
has already gone further and given to the field of a chart
invariably the color of green. The green should be medium
light with somewhat more yellow than blue, so that it will
photograph as well as gray. The advantage of the green is
that no confusion with black plotting or lettering is possible.
The rule may be extended to make it universal, even the
maps to be used in statistical reports being printed in green
outlines.
Red is a color to which the accountant is accustomed to
give a negative significance, and may well be used for con-
trast with black in curves and map-shadings, wherever it
709
710
CHARTS AND GRAPHS
applies to opposite data. Thus on a map the red should be
used for unfavorable conditions, and in curves for costs or
expenses, and the like. It is one of the^ great advantages of
the red that the corresponding data (and scales, if any) can
be typewritten in the same color without difficulty when the
typewriter is equipped with a bi-color ribbon. Brown, green,
and blue typewriter ribbons can also be secured, but require
special shifting of ribbons in the machine. Red and black
are the main colors used in the charting office.
Blue is taboo in chart-work and should rarely be used.
Its least fault is that it is hard on the eyes when used steadily
in the place of black. Its great fault is that it does not photo-
graph, and will therefore disappear if the chart is photostated
or blue-printed or photographed in other ways. In maps it
may be used for shading to indicate favorable conditions,
with the understanding that when photographed, these areas
will be white; in curve work it may be used for lines, or rulings
which are intended to disappear when photographed, either
for secrecy or to eliminate details useful only in plotting and
undesirable in reduced copies. When blue is used to disappear
.either in ink or typewriting, care should be taken to see that
it contains no red pigment, as this will defeat the purpose.
When a great deal of red is present, we have, of course, purple
—another color which should not be used, as it looks black
under most artificial light.
Yellow and green, are little-used colors, but sometimes '
serve in curve-work as secondaries to black and red, as for
example where it is desired to insert “quotas” or comparisons
in fainter colors, on the same chart with black and red curves.
These colors are also useful in map work, when areas are
shaded, as intermediates between the red and blue extremes —
the best sequence being red, orange, yellow, yellow-green,
and blue-green. The other colors, brown, pink, and the like,
are very little used, as they are not so distinctive as the simple
primary colors.
Appendix E
OPTICAL ILLUSIONS IN CHARTS
Mention has been made of the danger in bar-charts, and
area-charts, of making one area appear larger than another
merely through the use of more powerful shading. This
By permUsion of the publishers of the '*Book of Knowledge/ * 3 H est 45 ih Street, N, Y, City.
Fig. 498. Optical Illusions.
applies to colors as well as to the use of various grays and cross-
hatchings. But there are no hard-and-fast rules as to what
712
CHARTS AND GRAPHS
is a powerful shading, as each*case depends upon the surround-
ing colors or shades with which it is in contrast. The chart-
maker must in each case judge carefully »of the effects of his
shadings, and even if he cannot give equal emphasis tp all
parts of his chart, at least strive to avoid emphasizing the
unimportant and slighting the important parts of its message.
The accompanying illustrations show that, in the parti-
cular case presented, white is more powerful than black and
the white square appears larger than the black although the
black has a border or outline added to it. They also show
that a line may be made to look longer or shorter by the
direction of arrows attached to it, that hatchings make the
same rectangle look wider or narrower according to their
direction,, and also that bars hatched diagonally may be made
to appear crooked instead of kraight. These are but a few
of the minor optical illusions against which the maker of the
chart should be on guard.
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